The statemend does indeed hold in type H3; you can find the computation here: http://sage.lacim.uqam.ca/home/pub/15/. The computer is still running (and I don't know how long it might take) doing type H4.

So, if no Coxeter theoretic proof is known, there is hope to find one...

The statement does indeed hold in type H3; you can find the computation at http://sage.lacim.uqam.ca/home/pub/15/. The machine is still running (and I don't know how long it might take) doing type H4.

So, if no Coxeter theoretic proof is known, there is hope to find one...