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Can we fix one of these filter at the back of my PC? Would that actually work?

In order to prevent dust buildup inside the PC? Yes.

In order to meaningfully clean the room's air? Not on one PC, with any reasonable number of fans.

Some big radiator + unreasonably sized fans PC+filter build would be interesting at the very least

I dunno. With some people talking about how many CFM they can move with their fan system, maybe a filter or two would just add to things.