I used to have those type holders until the deterioration became too much. Here in CO the sun plays a strong role. I tried using old conveyor belt that I fabricated to fit the holder and pop riveted in. It worked fine and lasted quite awhile.
I finally came across a holder made
by D & S. It is essentially two solid rubber wheels facing one another, (like a baseball pitching machine), mounted to a bracket. The gap between the wheels has some spring to it and you push the foaming brush handle into the gap and the wheels spring together on the outside of the handle and capture it.
I have been using these for some years now with no problems.
Good luck