Keep right on to the end of the road is the message - however hard the journey. It is a good philosophy, but endurance alone is never enough. Birmingham City made it their anthem, and look what happened to them.
Nearly four months after his appointment, there is a very real danger of Walker's Everton going the same way. Merseyside's 'other' team have been in the top division for 40 years - longer than anyone bar Arsenal - but need to win at least one of their last two games to avoid relegation for the first time since the old king died.
'Other' team? It was not always thus. For most of their 106 years blue has held sway over red. Everton won the title before Liverpool, and had the edge until the Shankly era. Champions as recently as 1987, there is pride as well as prestige at stake.
The supporters are twitchy, and understandably so. Having to win one of the two remaining matches does not sound too bad, but blue faces blanche before the twin terrors of record book and fixture list. One shows just one win in Walker's 15 League games, the other that the make-or-break dates are away to Leeds United, where Everton have not won for 43 years, and at home to Wimbledon.
Leeds today probably represents their best chance. Wimbledon are on a roll, having seen off each of the top three, and as Peter Reid, one of Goodison's favourite sons, warned this week: 'They have no regard for reputations, and would revel in relegating a club like Everton.'
Walker had not expected it to come to this when he walked out on Norwich City amid acrimony which saw his new employers found guilty of poaching. In all, they were required to pay pounds 125,000 in fines and compensation.
The principal, who was an otherwise divided board's unanimous choice to succeed Howard Kendall, thought some hurried end-of-term cramming would see the old School of Science scrape a pass. Instead he found the whole syllabus in urgent need of revision.
Little more than a year ago, Kendall told me Everton were just a couple of players away from renewed prosperity. Sitting in the same chair, at the club's Bellfield training ground, Walker scoffed at the assessment.
'Last season they survived relegation by just four points, so they've not been going anywhere for quite a while now. To talk in terms of being not far away - two players off - is a bit of a statement, isn't it?'
His own view is that at least four new faces will be needed, a prolific striker among them, for substantial progress to be made. In the meantime, it was 'absolutely vital' that the relegation dogfight was won.
Winning the battle would be a lot easier with men who were fighting fit. It was Walker's greatest surprise, and disappointment, to find the players lacking in basic stamina. His Norwich team were hardly the SAS of the Premiership, but their leg and lung power was in a different league.
'Standards are not as high as they should be. They've been allowed to slip. People talk about resurrecting Everton and winning things again, but we've got to raise our levels of performance everywhere before we can do it.
'I don't think the players train hard enough. They think a little running session is hard work, but we're only talking about what was normal at Norwich. The Everton lads hadn't been used to doing any running. Their training routine was a few five-a-sides and a bit of head tennis. That was it.
'It shows. That's why they can't do certain things I want. They're not fit enough.'
Walker's football is more cerebral than physical, but he is a firm believer that the body has to be right to enable the mind to operate at optimum effectiveness.
'Everybody talks about how good the foreign players are - how great their touch is, how much skill they've got, their marvellous awareness - but very few seem to appreciate their fitness. You don't see an unfit European side. You don't see their players fat and bloated. They're all gaunt and look like athletes.
'That's why they're so good. They've got that mixture of technique and conditioning. Here, everybody likes a beer after the game, and some over-indulge.'
A more spartan regime would be beneficial for a variety of reasons, Walker subscribing to the military's theory that drill builds morale. 'I don't think there has been enough of that at this club. Hard work creates a camaraderie which in turn makes a group mentally tougher. They think: We'll get among this lot. We're better than them, we're stronger than them. I do think that attitude is important. I haven't found it here, but that will change.'
Enough of the work ethic. The coach who gave us sophistication on a shoestring was beginning to sound like Charles Atlas, or at least Charles Hughes, the high priest of the long ball. Sacrilege. The latest incumbent is in the very best traditions of Everton managers, to whom composure and precision has always been holy writ.
Walker inherited a team trying to play football the right way, and failing. 'I found that although they were renowned as a passing side, they were not that sharp on the ball. They were too laboured. They weren't clipping it around.
'That's one of the things I mean when I say the standard wasn't what it should be - one of the reasons why we're down near the bottom. We've worked on it, and they're getting better, but it's got to be a lot better still. That might be because the players are not good enough. Only time will tell.'
Everton had sold too many good ones, a cri de coeur from Norwich days. 'Peter Beardsley springs immediately to mind, Mike Newell hasn't done badly since he left, then there's Stuart McCall, Martin Keown - Graeme Sharp, even. These players still had plenty left. Give me any three of them and we'd be halfway up the table.
'Big clubs don't sell people like Beardsley, but we did. Now look at him - setting Newcastle alight and back in the England team. We should be gathering players of that quality, not letting them go.'
A start was made on redressing the balance by paying pounds 1.6m to rescue Anders Limpar from Arsenal's reserves, but another winger had served only to point up the most pressing problem. No matter how well the ball is crossed, Everton have no one capable of putting it in the net with an acceptable degree of regularity. In the last eight League games they have scored just three goals, all of them from Tony Cottee.
The Tom Thumb of the penalty area was again doing 'not badly' but, like Kendall before him, Walker would like more 'graft' from this sedentary six-yard poacher.
Graft, patience and nerve are Walker's managerial tenets. 'I want to play and move, like Norwich, but you've got to have an end product. I don't think we've got great footballers, the players are too much of a muchness, but we are fairly well off when it comes to the ones who can scrap it out.
'You wouldn't say Dave Watson was a classical centre-half. Dave is an up-and-at-'em boy, ideal for our present situation. Gary Ablett is capable of playing it that way, too, and Barry Horne and John Ebbrell are battlers. That's four, and you don't need a whole team of them.
'Yes, we've got difficult games, but so have Oldham, and I don't think Ipswich will be too chuffed about having to play Manchester United and Blackburn. It's too close for comfort all right, but I don't think we'll go down.'
The punters are not so sure. 'Mike mate, d'you realise how important this is to us?' enquires Anxious of Aintree. 'Hang on a minute, pal,' Walker retorts. 'However important it is to you, it's twice as important to me. We're talking about my bloody living here. Just have a bit of faith.'
The last two steps are big ones. Perseverance alone may not be enough.
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