Voters across the political spectrum said they’ve lied about their voting: 27% of Democrats acknowledged it, while 24% of Republicans and 20% of independents did so. The survey didn’t ask exactly how, why or to whom they’d lied.

This is what I was looking for. It’s not reliable data about which direction it may be influencing polling, but if a self-identified “democrat” is lying, presumably it is to conservative family or friends about conservative support (and vice versa). This would mean there is slightly more “shy” democrats than republicans, but with a very large “independent” black box.