Explicit basis conjecture for invariant differential forms and multiderivations of coincidental reflection groups

Let $W$ be a coincidental complex reflection group, let $S(V^*)$ be the symmetric algebra on the dual of its reflection representation, and choose basic invariants $f_1,\ldots,f_n$ and basic derivations $\theta_1,\ldots,\theta_n$. For $r=1,\ldots,n$, set

$$M_r=(S(V^*)\otimes\wedge V^*\otimes\wedge^r V)^W$$

and

$$R_r=\wedge_{S(V^*)^W}\{df_1,\ldots,df_{n-r}\}.$$

Here $\widetilde{\theta}_i$ denotes the differential operator induced by the derivation $\theta_i$ as defined in the preceding setup. Explicit basis conjecture. One may choose the basic invariants and basic derivations so that each $M_r$ is a free module over $R_r$, with basis

$$\widetilde{\theta}_{i_1}\cdots\widetilde{\theta}_{i_m}\left(\theta_{j_1}\wedge\cdots\wedge\theta_{j_r}\right)$$

for $0\leq m\leq r$, $1\leq i_1<\cdots<i_m\leq r$, and $1\leq j_1<\cdots<j_r\leq n$. For $r=0$, the basis consists of the element $1$. This conjecture strengthens the paper's main invariant-theoretic theorem by specifying an explicit basis; the paper states that it is verified for all remaining coincidental groups other than types $A$ and $G(d,1,n)$, while the general assertion as presented here has unresolved status.