Lexicographic homological linear quotients conjecture for powers of cover ideals

Let $G$ be a Cohen--Macaulay very well--covered graph with $2n$ vertices, and let $I(G)^\vee$ be its cover ideal. For each $k\geq 0$, let $\operatorname{HS}_k((I(G)^\vee)^\ell)$ be the $k$th homological shift ideal of the $\ell$th power, and order the variables lexicographically by

$$x_n>y_n>x_{n-1}>y_{n-1}>\dots>x_1>y_1.$$

Lexicographic homological quotients conjecture. For all $k\geq 0$ and all $\ell\geq 1$, the ideal $\operatorname{HS}_k((I(G)^\vee)^\ell)$ has linear quotients with respect to the lexicographic order induced by this variable order.

This is a refinement of the paper's broader conjecture that all powers of the cover ideal have homological linear quotients. The source gives no evidence that either conjecture has been resolved.