The flatness conjecture for universal quasiharmonics at constant parameter

Let $W$ be a real reflection group of rank $\ell$, with Coxeter number $h$ and basic invariant degrees $d_1,\dots,d_\ell$. Let $r$ satisfy $r\equiv d_i-1\pmod h$ for some $i\in\{1,\dots,\ell\}$, let $\mathcal{QH}_r$ be the degree-$r$ component of the universal quasiharmonics, and let $V^{(r)}$ be its distinguished copy of $\mathbb{C}[c]\otimes V$. Write $\mathcal{QH}_r^{(r/h)}$ for the degree-$r$ quasiharmonics at the constant parameter $c=r/h$, and $\operatorname{ev}_{r/h}\mathcal{QH}_r$ for specialization at that parameter. Flatness conjecture. If

$$c=\operatorname{const.}=r/h,$$

then

$$\mathcal{QH}_r^{(r/h)}=\operatorname{ev}_{r/h}\mathcal{QH}_r.$$

In particular, if $V_r$ denotes the known subspace of singular vectors in $M_r^{(r/h)}(\mathbf{1})$, then $V_r=\operatorname{ev}_{r/h}V^{(r)}$. The conjecture is proved for dihedral groups, and computations are given for symmetric groups; it remains open in general.