Which Piece Least Needs A Power Cord Upgrade?

I agree, listen and hear if there is a major upgrade. Better yet, leave the room and have a friend switch out the power cords and then have a listen, not knowing which one is in place.
Yup many will balk at the thought of a blind test, but with power cords you really need to take the "eye candy" out of the picture.
You have a very nice system Zar, and your money is well spent on true audio upgrades rather than the visual appeal of "fat" power cords.

You need to listen & make that decision for yourself. Just because a cord or cable is beefier, etc, doesn't mean its better. The same cord can sound different on the other pieces in your system.

I would also suggest running straight to the wall in your evaluation to make sure the strip isn't affecting the sound.

I see where you installed dedicated lines and suggest you look into changing the wall outlets if you haven't already done so. I found both tailoring the wall outlets and PC's to really help create the kind of presentation I was looking for.

From looking at your pics, the hard surface of the fridge in the corner needs to be tamed. You can use some type of room treatment but first you should pull the spkrs out into the room to hear the difference. Moving spkrs. is the most cost effective and easiest tweak you can perform.

Check out Morrow Audio MAP 2 power cords (<$200), replace supplied plugs with Wattgate 330i/350i