Lower bound for sdepth of quotient modules by monomial ideals

Determine whether the inequality sdepth(S/I) ≥ depth(S/I) − 1 holds for every monomial ideal I ⊂ S = K[x1, x2, …, xn].

Background

After the general Stanley conjecture failed for quotient modules J/I, a refined bound for the Stanley depth of S/I was proposed. Verifying whether sdepth(S/I) is always at least depth(S/I) minus one would provide a near-Stanley-type inequality for quotients by monomial ideals, partially salvaging the original conjecture in an important class of modules.