It's a sign of a high EQ to have a neutral response to being told 'your IQ is x'. It's actually just another part of the battery of psychological tests.
That could easily be true without people lying (though probably not). If you take a test and score a 99, you aren't exactly about to run around saying "Hey everyone, I'm very slightly below average according to an IQ test!"
There is a lot of selection bias there. 9 out of 10 people I see day-to-day make more than twice the per-capita median income. That isn't evidence that the median income is incorrectly computed.
The kind of people who go around talking / bragging about their IQs are likely to overlap with that set of people that score well on formal cognitive tests, IQ or otherwise. (See, the entire membership of "Mensa", for example.)
(And if a person who was very into tests and intelligence also scored 95, you'd probably not hear a peep out of them - it would misalign with their self image.)
Finally, the typical person scoring 80-85 in IQ is not likely to be the kind of person who enjoys taking such tests, or considers their IQ or their intellectual ability as an important aspect of their personal identity.
I got a bit of confirmation of that while taking an IQ test at a Mensa meeting. The proctor mentioned that generally something like 2/3 of the people who took the test there scored well enough to get in.
