Existence of nonzero higher differentials in the local defect sequence

Determine whether there exists a meet functor arising from a finite graded poset whose local defect spectral sequence has a nonzero differential $d^r$ for some $r\ge2$.

Background

The paper bounds the possible columns in the spectral sequence computing the derived colimit of the local defect module and identifies conditions under which the spectral sequence degenerates at the second page. In all examples treated, either the first window is of unit size or the relevant fibre is sufficiently simple, so higher differentials vanish.

The authors explicitly state that no meet functor treated in the paper exhibits a nonzero higher differential and that no example is known in which such a differential occurs. The existence of such an example remains unresolved and is connected to the possible non-formality of the defect module over the relevant incidence algebra.