Carpet moths in synthetic carpet?

Can moths (the teeny-weeny, destructive kind) lay their eggs in synthetic carpeting? Or what I mean is, can the larvae survive if they do? Because I can’t figure out where else they could be coming from.

The little buggers are way too fast to vacuum up out of the air. I am at my wits’ end.

It seems to be a banner year for bugs in general. Well, except bees, alas.

Mine were nesting on an acrylic jacket lining and also in a box of #10 paper envelopes. Got rid of them for a couple of weeks but now one flits around about every evening. Grab of hand has worked for me, or a snout full of aerosol bug bomb.