Quaternionic Haiman conjecture for zero-fiber rings

Let $W$ be an irreducible quaternionic reflection group containing $N$ reflections and acting in the $n$-dimensional quaternionic vector space $V$. Put

$$g=2N/n.$$

The zero-fiber ring of $W$ is the ring of functions on the scheme-theoretic fiber over zero of the quotient map from $V$ to $V/W$.

Quaternionic Haiman conjecture. There is a $(g+1)^n$-dimensional quotient ring of the zero-fiber ring of $W$.

This is proposed as a quaternionic analogue of Haiman's conjecture for real reflection groups. The paper proves the asserted quotient dimension for irreducible imprimitive quaternionic reflection groups of rank at least $3$ and for quaternionifications of irreducible complex reflection groups, while the rank-one zero-fiber ring is shown to have dimension exactly $g+1=2|W|-1$. The general conjecture is not resolved by the supplied text.