Negative extended-Shi conjectures for negative powers of the nabla operator

Let $D^{mn-1}(n)$ be the dilated simplex associated with bounded chambers of the $m$-extended Shi arrangement, and let ${\sf shi}^m(A)$ and ${\sf ish}^{-1}(A)$ be the corresponding statistics on its alcoves. Define

$${\sf Shi}^{-m}(n;q,t):=\sum_{A\subseteq D^{mn-1}(n)}q^{{\sf ish}^{-1}(A)}t^{(mn-2)(n-1)/2-{\sf shi}^m(A)}.$$

Let ${\sf Shi}^{-m}_+(n;q,t)$ denote the same sum over alcoves whose inverses lie in the dominant cone $C_\circ$. Write $[r]_q$ for the $q$-integer and ${a\brack b}_q$ for the $q$-binomial coefficient. Negative extended-Shi conjectures. The following assertions hold:

1. $(-1)^{n-1}{\sf Shi}^{-m}(n;1/q,1/t)/(q^{n-1}t^{n-1})$ is the Hilbert series of $\nabla^{-m}(e_n)$.
2. $q^{(mn-2)(n-1)/2}{\sf Shi}^{-m}(n;q,1/q)=[mn-1]_q^{n-1}$.
3. $(-1)^{n-1}{\sf Shi}^{-m}_+(n;1/q,1/t)/(q^{n-1}t^{n-1})$ is the Hilbert series for the sign-isotypic component of $\nabla^{-m}(e_n)$.
4. $q^{(mn-2)(n-1)/2}{\sf Shi}^{-m}_+(n;q,1/q)=\frac{1}{[n]_q}{(m+1)n-2\brack n-1}_q$.

The conjectures propose a combinatorial interpretation of negative powers of the nabla operator using bounded chambers of extended Shi arrangements, extending the positive-power picture. The source explicitly says that no combinatorial conjectures for negative powers had previously been available and gives no resolution evidence in the provided text.