Lattice Stanley projective dimension inequalities conjecture

Let $L$ be a finite atomistic lattice. Choose a monomial ideal $I\subset S={\mathbb K}[X_1,\ldots,X_n]$ whose lcm-lattice is isomorphic to $L$, and define

$$\operatorname{pdim}_1 L:=\operatorname{pdim} I,\qquad \operatorname{pdim}_2 L:=\operatorname{pdim} S/I,$$ $$\operatorname{spdim}_1 L:=\operatorname{spdim} I,\qquad \operatorname{spdim}_2 L:=\operatorname{spdim} S/I.$$

Here $\operatorname{spdim}_S M:=n-\operatorname{sdepth}_S M$.

Lattice Stanley projective dimension conjecture. For all finite lattices $L$, the following inequalities hold:

1. $\operatorname{spdim}_1 L\leq\operatorname{pdim}_1 L$;
2. $\operatorname{spdim}_2 L\leq\operatorname{pdim}_2 L$;
3. $\operatorname{spdim}_1 L\leq\operatorname{spdim}_2 L-1$.

These inequalities are lattice-theoretic formulations of conjectures concerning Stanley depth for monomial ideals and their quotients. The source presents them as a conjecture from the cited earlier work; no resolution is supplied here.