There are, of course, several versions of the AFR. If the opponent is advocating some kind of reductionism, then all the difficulties in providing a naturalistic account of intentionality come into play.
If the opponent is a non-reductivist, they are basically giving up on the idea of coming up with an analysis of intentionality in physical terms. Instead, they guarantee intentionality through a supervenience relation. Now, this doesn't explain intentionality, because we apparently are not told why this relation exists. It strikes me as a kind of necessity-of-the-gaps response, a kind of mystery maneuver. To the question "why are there genuine intentional states, and real conscious persons, as opposed to just behavior that looks intentional or conscious" the nonreductivist really isn't providing any answer.
I think a physicalist has to be bothered by the fact that they are positing a supervenience relation as an ultimate brute fact, even though it isn't a physical brute fact. So, I wanted to pose some problems for the ontological status of supervenience itself, which strikes me as questionable.
But, there has always been a recognized problem for the nonreductivist in the area of mental causation. If they physical is causally closed, that means nothing non-physical can cause anything. Now cause-and-effect relations between mental states seem to me to be necessary for the possibility of science. Einstein has to do the math, and his doing the math has to cause him to propose his theory. Without mental causation, it cannot be true of us that we literally add, subtract, multiply, and divide numbers, much less analyze Maxwell's equations. And yet physical states, not mental states, do the causal work in the materialist's universe.