This is most likely an urban myth!

There is an idea that if you are searching again the website should put up the prices to cause you to panic and buy now. But there is also an idea to the contrary effect that if the passenger has not bought before now the price may have been too expensive.

It is possible that some airlines might have experimented with this technology to see if it increases revenue.

But I doubt that BA has and even if they did consider there would be no guarantee that it would be a sound business decision.