The graded-piece characterization conjecture for $R_{i,\mathbf{h}}$

Let $D$, $R_{i,\mathbf{h}}$, $S_{i,\mathbf{h}}$, and the weight $\mathrm{wt}(\mathbf{h})$ be as in the preceding notation. For $j\geq 0$, suppose that $i=j+D$, and define

$$\mathcal{H}_j:=\{\mathbf{h}'\mid i-\mathrm{wt}(\mathbf{h}')\in k\mathbb{N},\ \mathrm{wt}(\mathbf{h}')<d-j,\ \mathrm{wt}(\mathbf{h}')\leq (k-1)(n+1)\}.$$

Graded-piece characterization conjecture. One has $R_{i,\mathbf{h}}\neq 0$ if and only if $\mathbf{h}\in\mathcal{H}_j$; moreover, for $\mathbf{h}\in\mathcal{H}_j$,

$$\dim_{\mathbb{C}}R_{i,\mathbf{h}}=\dim_{\mathbb{C}}(S_{i,\mathbf{h}})-{{n+j}\choose{n}}.$$

The statement is presented as a consequence that would follow from the preceding conjecture on the maps $\mu_{i,\mathbf{h}}$, so its validity is likewise conditional and open in the supplied text.