Modular Betti-shift stability conjecture for symmetric-group cofixed spaces

Let $p$ be a prime. For each $r\geq 1$, define

$$R_r:=\mathbf{k}[e_1,\ldots,e_r],\qquad M_r:=\mathbf{k}[x_1,\ldots,x_r]_{\mathfrak{S}_r}.$$

For an integer $i\geq 0$, let $A_i^R(M)$ be the multiset in which $j$ occurs with multiplicity $\beta_{i,j}^R(M)$, for a graded module $M$ over a graded ring $R$. Let $m,n$ satisfy

$$kp\leq m,n<(k+1)p$$

for some integer $k$, and let $\mathbf{k}=\mathbb{F}_p$ or $\mathbf{k}=\mathbb{Z}_{(p)}$. Modular Betti-shift stability conjecture. For each $i\geq 0$, the multisets $A_i^{R_n}(M_n)$ and $A_i^{R_m}(M_m)$ are equal after reducing every element modulo $p$. This conjectures that the degree shifts in each homological degree of the minimal resolutions depend, modulo $p$, only on the interval between consecutive multiples of $p$ containing the number of variables. The phenomenon is known in the smaller ranges discussed in the source, while its persistence for all such $m,n$ is left open.