The injectivity-surjectivity conjecture for the maps $\mu_{i,\mathbf{h}}$

Let $\mu_{i,\mathbf{h}}$ be the maps defined in Definition, indexed by $i$ and a vector $\mathbf{h}$, and let $\mathrm{wt}(\mathbf{h})$ denote the weight of $\mathbf{h}$. Let $k$, $d$, and $i$ be the parameters appearing in the definition of these maps. Injectivity-surjectivity conjecture. In the same notation, $\mu_{i,\mathbf{h}}$ is injective when

$$\mathrm{wt}(\mathbf{h})\leq (k-1)(d-i),$$

and it is surjective when

$$\mathrm{wt}(\mathbf{h})\geq (k-1)(d-i).$$

This conjecture is proposed as a generalization of the corresponding behavior established in the $k=2$ setting and is intended to yield further descriptions of the graded pieces of the algebra for $k>2$.