Isn’t the point that he denied he’d bought the house, and his colleague remembered that he had.
That was probably the opportunity for the “quiet word”, the quiet word was rebuffed.
I know a bent copper who was kicked out of the Met for fraud. He also rebuffed the invitation for a quiet word with a denial, being a grade one asshat he went a little further with “I didn’t do it, and you can’t prove I did”.
This was about 15 years ago, if it had happened now, he’d he inside I’m pretty sure. He was just kicked out by a tribunal.
It appears the reason the other chap went “private detective” was because the accused rebuffed and denied instead of accepting a quiet word.