Algorithmic Lyubeznik–Barile–Macchia conjecture

Let $I$ be a monomial ideal, let $\mathcal{F}_L$ be a Lyubeznik resolution, and let Algorithm $1$ be the trimming algorithm defined in the source. The resulting complex is compared with Barile–Macchia resolutions. Algorithmic Lyubeznik–Barile–Macchia conjecture. Applying Algorithm $1$ to a Lyubeznik resolution produces a Barile–Macchia resolution up to isomorphisms. The conjecture would explain why the algorithm systematically yields the rank inequality from the preceding conjecture and would extend the constructions known for certain edge ideals; the source gives this as an open direction rather than a proved result.