Might be mites, but looks more like cicadas to me. I have pigeons peas and their leaves look like that and that's from little green cicadas ( I forgot the right name). Or it might be thrips, but those you should easily see as black 1-2 mm skinny insects, they hang around the leaves. Go on a insect hunt, only that way you can id what's going on in there.

It does look like spider mite damage. The TINY bugs would be on the bottom of the leaves, with a lot of spider webbing. Try gently wiping the leaves with a q-tip or cotton ball damp with rubbing alcohol.

Is the soil staying too soggy? That could make the plant unhealthy overall, which would then make it more susceptible to pests in general.

Do the white specks, which you describe like a dust, wipe off or are they a part of the leaf tissue? If the latter, I'd suggest a very bad case of spider mites. They are so small that they are nearly invisible, but can be seen crawling around (usually on the back of the leaf) with the aid of a magnifying glass.

Other possibilities would be thrips or tiny leaf hoppers....which you would surely notice.

part of this.. involves you taking a very old.. aged.. ANNUAL ... and trying to winter it over ...

a young vigorous plant .. can take a lot more abuse .. rather than grandpa ...

whatever the problem.. in a small lean-to greenhouse.. if it were mine.. this thing would be chucked over the neighbors fence.. and the rest of the clean plants would be treated ... if possible, on a food crop ..

this plant should have been prepped to bring indoors..

your pestilence.. has the potential to wipe out everything in your greenhouse.. how long are you going to fool around with it???

in other words.. IMHO.. your focus is the rest of the plants.. not this plant ....

ken

ps: this is where you tell us its the only known pepper of this type in the universe.. and you just cant live w/o it ...

Thank you everyone! I think it is spider mites. I haven't been able to find them on all the leaves, but some of them seem to have tiny creatures on the underside. I appreciate all the helpful suggestions!