Is carpet on wall behind speakers counter productive?

If you have some boards you could put up in front of the carpet, temporarily, even to cover a fair portion or half the space, you could quickly and easily observe what happens to the sound. Many people do dampen that front wall, but carpet is a clumsy and awkward absorber.

"If you stick carpet directly to a wall, it isn't thick enough to affect more than the very highest frequencies, thus throwing your whole room out of balance! The lows and mids will continue to bounce around as before, and only the extreme highs will be soaked, leaving you with a boxy, dull-sounding room. However, if you hang the carpet a distance from the wall, it will start to have a more beneficial effect. Providing the carpet is porous (not one with a sealed rubber backing), it can be used to cover a cheap absorbing material such as medium density Rockwool (mineral wool) slab 50 to 100 mm thick, to create an effective mid/high absorber with an attractive cosmetic finish that also prevents the mineral wool fibres from escaping."
https://www.soundonsound.com/sound-advice/q-can-use-carpet-acoustic-treatment

Probably not. See my system. A deflector behind the speakers is a very common acoustical treatment. It improved my system sound quite a bit.

Not a problem. I have set up systems with a carpeted front walls. Saves you from having to find reflections and dampen them. The carpet is not as good as 4 inch acoustic foam but if you are not using dipoles you probably do not need it. Leave the carpet up!