Well therein lies the problem: perceptual hashes don't produce an exact result. You need to compare something like the hamming distance (as the article mentions) of each hash to decide if it's a match.

Is it possible to perform private set intersection where the comparison is inexact? I.e., if you have two cryptographic hashes, private set intersection is well understood. Can you do the same if the hashes are close, but not exactly equal?

If the answer is yes, that could mean you would be able to derive the perceptual hashes of the CSAM, since you're able to find values close to the original and test how far you can drift from it before there's no longer a match.