Generalized Lyubeznik-to-Barile–Macchia resolution conjecture

Let $I$ be a monomial ideal. A minimal generalized Lyubeznik resolution and a minimal generalized Barile–Macchia resolution are resolutions of $I$ with the indicated generalized constructions and minimality properties. Generalized Lyubeznik-to-Barile–Macchia conjecture. If $I$ has a minimal generalized Lyubeznik resolution, then $I$ also has a minimal generalized Barile–Macchia resolution. The conjecture proposes that the existence of the former resolution guarantees the existence of the latter; the supplied text does not indicate whether it has been resolved.