Projective-dimension formula for t-clique ideals

Determine whether t-clique ideals satisfy the equality pd(R/I(𝓒_t(G)))=bight(I(𝓒_t(G))) for all relevant graphs G and integers t≥2.

Background

The authors explain that t-clique ideals satisfy the linear-resolution condition associated with complements of chordal graphs, while the regularity formula involving the induced matching number fails in general.

They explicitly state that they do not know whether the projective dimension equals the big height for t-clique ideals. The question concerns condition (ii) of the three-part generalization proposed in the concluding remarks.