Shellability and Hilbert-series conjecture for principal components of determinantal jet schemes

Let $\mathscr{L}^{m,n}_{r,k}$ be the jet scheme under consideration, let $\tilde{I^{m,n}_{r,k}}$ be the ideal corresponding to its principal component, and let the corresponding Stanley–Reisner abstract simplicial complex be the complex associated to the initial ideal of \tilde{I^{m,n}_{r,k}. The Hilbert series of $\mathscr{L}^{m,n}_{r}$ denotes the Hilbert series of the corresponding base determinantal variety.

Shellability and Hilbert-series conjecture. The abstract simplicial complex corresponding to $\tilde{I^{m,n}_{r,k}}$ is shellable, and the Hilbert series of the principal component of $\mathscr{L}^{m,n}_{r,k}$ is exactly the $(k+1)$-st power of the Hilbert series of $\mathscr{L}^{m,n}_{r}$:

$$H_{\operatorname{Prin}(\mathscr{L}^{m,n}_{r,k})}(z)=\left(H_{\mathscr{L}^{m,n}_{r}}(z)\right)^{k+1}.$$

The conjecture extends the explicitly verified cases discussed in the paper, including the cases $\mathscr{L}^{m,n}_{2,1}$ and $\mathscr{L}^{3,n}_{3,1}$. The paper states that the principal component exists in general and gives an explicit description, but the asserted shellability and Hilbert-series formula are presented as conjectural in the general case.