Pardue's characteristic-independence conjecture for p-Borel-fixed ideals

Let $S$ be the polynomial ring from the paper, let $p$ be a prime number, and let $I$ be a $p$-Borel-fixed monomial $S$-ideal. For every field $\ell$, form the base change $I(S\otimes_{\mathbb Z}\ell)$.

Pardue's conjecture. The $\mathbb{N}$-graded Betti table of $I(S\otimes_{\mathbb Z}\ell)$ is independent of $\operatorname{char}\ell$ (equivalently, of $\ell$) for all fields $\ell$, including fields of arbitrary characteristic.

The conjecture concerns whether $p$-Borel-fixed ideals have characteristic-independent graded Betti tables, extending the corresponding fact for $0$-Borel-fixed ideals. The source reports evidence for the conjecture but does not establish it in general.