The Liberal Democrats are trying to woo an estimated 17,000 undecided former Conservative supporters in the 96,000-voter constituency. About one in four of these would have to turn their backs on the Tories to ensure a Liberal Democrat victory.
But if the bulk defect to the Liberal Democrats' David Chidgey rather than stay at home, the party could win by up to 15,000, not far off the 17,702 majority bequeathed by the late Tory MP, Stephen Milligan, at the general election.
But for Labour the prospects are of a respectable but unexciting performance that is not expected to outstrip its 21 per cent share of the vote at the 1992 elections.
That will be embarrassing for a party that has spent thousands of pounds on a campaign that began at the end of February, involving a clutch of heavyweight national officials and constant visits from senior politicians, including the late Labour leader, John Smith.
Marilyn Birks, the Labour candidate, declared at her news conference yesterday that 'local election results and opinion polls are showing us that Labour is firmly in second place' - the first time she has conceded she cannot win.
The Labour high command has repeatedly insisted that to concede that south of England seats such as this are unwinnable would be political dynamite in the hands of the Tories. Tony Blair, the most likely future Labour leader, echoed that view during his third by- election visit yesterday. 'There's no point in going into a by-election unless you're going to win.'
But in the 5 May council contests, Labour and the Tories each captured 10,000 votes on a 53 per cent turn-out, compared with the Liberal Democrats' 20,000.
A crucial Tory objective in the next 36 hours is to counter 'voter fatigue' after these elections, which is one reason why turnout may fall from about 70 per cent to about 65 per cent. A key message is that former Conservatives must now return to the fold, but the announcement of 136 job losses at Pirelli, the cable maker and leading local employer, will not help.
The Tories were given a boost yesterday as Lord Archer, the former deputy party chairman, accompanied Stephen Reid, the Conservative candidate, on walkabouts in Woolston, a traditionally Labour area near Southampton, and in the town of Eastleigh.
Constantly recognised for his novel-writing, Lord Archer's visit is likely to have eclipsed Mr Blair's meeting with senior police officers, and even that of Paddy Ashdown, the Liberal Democrat leader, whom the Liberal Democrat-controlled Eastleigh borough council insisted should park his helicopter at an outlying recreation ground.
But as the by-election winds up with visits today from Douglas Hurd, the Foreign Secretary, Gordon Brown, shadow Chancellor, and a final rally with Mr Ashdown, there seems little doubt about the Liberal Democrats' appeal to disenchanted Tories.
Scores on the doors, page 19