While spokesmen on both sides of the Atlantic deny any such plans on the part of the star investors, who include Arnold Schwarzenegger, Bruce Willis and Sylvester Stallone, the shares fell on Friday to just over $20, one-third below the peak of $32 they touched just a month ago.
Wall Street analysts point out that the fall happened as the stars have come to the end of a "lock-in" period during which their shares were untradeable. Now they are free to sell, and some of them would stand to make as much as $50m if they did.
Without the big-name owners, the experts warn, the brand would lose the glamour that is its unique strength. "It would send a bad signal - celebrity support is the key competitive advantage of the company," said one analyst.
Matthew Freud, Planet Hollywood's London spokesman, said speculation about the celebrities selling is "inevitable when you get the lifting of the restrictions". But he added: "Those restrictions have been lifted for a week and there is no sign of a celebrity looking to sell shares."
With the shares sliding, however, stock market experts used to poring over financial tables now find themselves scrutinising Hollywood gossip sheets and the glossy press to keep an eye on Planet Hollywood's glitzy shareholders.
It is the affairs of Mr Stallone that have aroused most interest, since the star of the Rocky and Rambo movies was recently involved in a protracted and bitter legal battle with his former financial manager.
A long article in New York Magazine last week described his dispute with his former stepfather, Anthony Filiti, who managed the star's money for five years and, says Stallone, cost him millions. The matter ended up in court when Mr Filiti filed law suits against the star and his new managers and Stallone counter-sued. In the end, they settled on undisclosed terms.
Mr Filiti's place in Stallone's affairs has been taken by Robert Earl, Planet Hollywood's billionaire-founder and chairman, who has lent his considerable talent to rearranging the actor's finances.
Last week in Los Angeles, Mr Earl insisted the stars were standing by Planet Hollywood: "100 per cent, there are no sellers." But he accepted that without the "lock-in", the company was vulnerable to the vagaries of the market. Removing the stars' handcuffs, he said, made Planet Hollywood like any other company, where "someone has stock which is stock with no restrictions".
In London, Mr Freud pointed out that this week will see the gala opening of another European branch, this time in Amsterdam. Ten or 12 stars will be there - a figure, he claimed, bettered only by the Oscar ceremony.Reuse content