I ended up ripping it out because it took me so long to finish the speakers that my kids were past finger-poking age, but when I made the grills for my RBR speakers, I stabled metal window screen to a recessed section of the grill frame prior to covering them with grill fabric. I never listened with and without the screen, but I bet you could stick a tweeter out your window and do some measurements.😀 I doubt the screen would have any impact, and you could glue some thick wool to the screen around the tweeter to help minimize diffraction issues from the grill frame.
There's a huge damping panel behind that curtain, that's what she used to climb up, next she slides the curtain aside and climbs on top of the speaker. 😱
