Speculation is all well and good, but I wanted data. I selected an OTB quick tournament from February and compared final scores with initial quick and regular ratings. After removing any previously unrated players, (surprisingly, everyone who previously had a regular rating also previously had a quick rating) the scatterplots looked like this:
The regular ratings did appear to be slightly better predictors in this case than the quick ratings.
Of course, this is a rather small sample size (27 players in one 4-round tournament) and they aren't independent since the players were playing each other. By using score as a measure I'm ignoring strength of opponents. Also, some players received one half or full point bye, and this of course affected their final scores.
Then I took a blitz tournament at the same day and location. (Mostly chosen so I wouldn't have to re-look up the regular ratings of most of the players; there was a lot of overlap in the player pool between these two events.) After removing players with no previous blitz rating, and also removing a withdrawal and a late entry, I got this:
It's a lot harder to say by looking whether the correlation is better between the blitz and the regular. Again, it's just one small tournament (10 rounds, and only 16 players after removing the ones without a previous blitz rating.)