Lyubeznik–Barile–Macchia rank conjecture for monomial ideals

Let $R$ be a polynomial ring over a field $\boldsymbol{k}$, and let $I\subseteq R$ be a monomial ideal. Let $\mathcal{F}_L$ be a Lyubeznik resolution of $R/I$. A Barile–Macchia resolution $\mathcal{F}_B$ of $I$ is a free resolution obtained by the Barile–Macchia construction. Lyubeznik–Barile–Macchia rank conjecture. There exists a Barile–Macchia resolution $\mathcal{F}_B$ such that

$$\operatorname{rank}(\mathcal{F}_B)_i\leq \operatorname{rank}(\mathcal{F}_L)_i$$

for each $i\in\mathbb{Z}$. This conjecture concerns whether Barile–Macchia resolutions can always be chosen, possibly using a different ordering, to be no larger in every homological degree than a given Lyubeznik resolution; the source presents it as motivated by examples and as an open question.