Haiman's trivial-representation kernel conjecture

Let $W$ be a real reflection group and consider the natural surjection of bigraded $W$-modules

$$D\!R^{(m)}(W)\otimes\epsilon\twoheadrightarrow\operatorname{gr}(L^{(m)}).$$

Haiman's kernel conjecture. The kernel of this surjection does not contain a copy of the trivial representation. For $m=1$, the source attributes the conjecture to Haiman; the source gives no resolution for the stated generality.