Bounded comatching number and Lerayness

Determine whether there exists an absolute constant d > 0 such that, for every set family F with comatching number at most 2, the nerve complex N(F) is d-Leray.

Background

The paper studies the relationship between the comatching number of a set family and structural properties of its nerve complex, including d-collapsibility and d-Lerayness. Although bounded comatching number yields strong colorful Helly and fractional Helly consequences, Proposition 4.2 shows that comatching number bounded by 2d does not force (3d − 1)-Lerayness or (3d − 1)-collapsibility. The authors therefore ask whether some absolute, possibly larger, dimension bound can nevertheless be guaranteed in the special case of comatching number at most 2.