The Scarf resolution conjecture for powers of extremal ideals

For an integer $q\geq 1$, let $\mathcal{S}_q=\mathsf{k}[x_A\colon \emptyset\neq A\subseteq [q]]$, and let $\mathcal{E}_q$ be the ideal generated by

$$\epsilon_i=\prod_{i\in A\subseteq [q]}x_A\qquad (i\in[q]).$$

For $r\geq 1$, write $\mathcal{E}_q^r=(\mathcal{E}_q)^r$ and let $\mathbb{S}(\mathcal{E}_q^r)$ denote its Scarf simplicial complex. The Scarf resolution conjecture. For all integers $r,q\geq 1$, $\mathbb{S}(\mathcal{E}_q^r)$ supports a minimal free resolution of $\mathcal{E}_q^r$.

A Scarf resolution is the smallest possible cellular resolution in this setting, and would give an explicit minimal free resolution for every power of every extremal ideal. The conjecture was previously speculated in the cited work and is presented here as a guiding conjecture supported by the paper's additional evidence and results.