The total degree bounds conjecture for pseudo-reflection groups

Let $G$ be a pseudo-reflection group. Write $\mathcal{SH}_G^{i,k}$ for the component of $x$-degree $i$ and $\theta$-degree $k$, let $\Delta_G$ be the discriminant, and let $\mathcal{SH}_G^{\det}$ denote the determinant-isotypic component. Total degree bounds conjecture.

$$\bigoplus_{i+k=d}\mathcal{SH}_G^{i,k}\neq 0\quad\Leftrightarrow\quad 0\leq d\leq\deg\Delta_G.$$

Moreover,

$$\bigoplus_{i+k=\deg\Delta_V}\mathcal{SH}_G^{i,k}\subset\mathcal{SH}_G^{\det}.$$

The statement is known in the cases covered by the preceding bi-degree bounds results and has been proved for the relevant infinite families except for possible exceptional parameter cases; the general pseudo-reflection-group claim remains open.