Use a small, flat and thin screwdriver to pry the joystick potentiometer slots open. Push the slots down about 80°. My screwdriver was was about 1,5mm in width and very thin.

Remove the potentiometer for the stick you have a problem with (remember the controller now might be mirrored)

Use a q-tip and electronic cleaner to clean the carbon track (everything on the inside of the green plastic) thoroughly.

Let it dry and do it again, with a new swab. I recommend doing this multiple times. It was first on my fifth attempt it worked without flaw.

Be careful not to let residue from the Q-tips get stuck.