No offense but this shouldn't even be a question. If it IS A DRUG, then YES, it is inevitable you will gain a tolerance to EVERY single side effect that drug creates. This is brain chemistry 101. You push a chemical this way, your brain fights to push it back the other way.
So yes, the anxiolytic effects will slowly dissociate with extended use.
However, your body does still stablize if you do not raise the dose. So you will gain tolerance to a certain extent, but then will find to a lower degree, but more consistently it will stave off anxiety.
It won't be as much as initially, but it will be more consistent. When you raise the dose again, tolerance develops again, you stablize, then raise the therapeutic floor another milimeter or so. Its how it goes will all drugs. Don't get depressed about it or anything, because it will still work. And even after I developed a tolerance to lyrica I noticed it still worked fairly well for anxiety, nothing like the first day I took it where I was starting conversations with every single person I saw, but more like when I needed to start a conversation it was nothing like being off the lyrica. Way easier, was more social, and after a certain point it became a new norm for my behavoir. At least till you stop the lyrica lol.
But shorter answer is yes.