One piece of common wisdom on LW is that if you expect that receiving a piece of information will make you update your beliefs in a certain direction, you might as well update already instead of waiting.

It's more like, if you expect (in the statistical sense) that you will rationally update your beliefs in some direction upon receiving some piece of evidence, then your current probability assignments are incoherent, and you should update on pain of irrationality. It's not just that you might as well update now instead of waiting. But this only applies if your expected future update is one that you rationally endorse. If you know that your future update will be irrational, that it is not going to be the appropriate response to the evidence presented, then your failure to update right now is not necessarily irrational. The proof of incoherence does not go through in this case.