The differential operator conjecture for pseudo-reflection groups

Let $G$ be a pseudo-reflection group, with $\theta$-degree $r$ as in the differential-operator identity

$$\mathcal{SH}_G=K[\partial_{x_1},\ldots,\partial_{x_n}]\mathcal{SH}_G^{\det}=\{\partial_g\mathrm{d}_{i_1}\cdots\mathrm{d}_{i_k}\Delta_G:g\in K[x_1,\ldots,x_n],\ i_j\in [r]\}.$$

Differential operator conjecture. If $G=G(m,1,n)$, then this identity holds. The conjecture extends the proved cases for rank at most two and the real pseudo-reflection groups; computational evidence shows that the identity does not hold for some other families, while it has been verified in several additional small cases.