I also know of cases (not myself) where HMRC asked for a logbook and then required confirmation from each airport logged, where the pilot was claiming business travel.
This is to be expected - it's the same with cars. Fraudulent business travel claims are very common, and in the worst cases the customer gets contacted to verify the visit. I've had quite a few such requests over the years in my business. Usually they come from the companies for which the sales rep works/worked but in the came of a sole trader / proprietor doing the alleged crookery the check will be done by HMRC.
I have never come across a pilot forging his logbook (would I know??) but when I used to rent the plane out I had a couple of people (instructors!) forging the fuel usage to get a lower invoice, by resetting the FOB setting on the fuel flowmeter. This is a lot worse...