Stanley inequality for monomial ideals

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

Background

The Stanley inequality asserts that the Stanley depth of a multigraded module M is at least its depth. Although this inequality (the Stanley conjecture) was disproved in general for quotients J/I of monomial ideals, the special case where M is a monomial ideal I itself remains unresolved. Determining the validity of sdepth(I) ≥ depth(I) for all monomial ideals would settle the remaining case of the classical Stanley conjecture in the monomial setting.