Might the M-sigma relation extend to galaxy clusters and higher?

I have a question which I can't resolve online and
perhaps you might answer if you have the time.

I'm familiar with the M-sigma relation for supermassive black
holes and the motions of stars in elliptical galaxies and spiral
bulges. Which goes roughly like:

Mbh varies with (more or less) the fourth power of velocity
of stars.

It occurs to me to ask: Does something roughly
similar exist for superclusters of galaxies, where clusters
and individual galaxies are proxies for stars in the relation's
original loci? I ask because of the prominence of super-
massive BHs in clusters and superclusters. Not to mention
dark matter.
Thanks,
Gene