The lattice formulation of Stanley's conjecture

Let $L$ be a finite semilattice. Choose an ideal $I\subset S={\mathbb K}[X_1,\dotsc,X_n]$ such that $L_I\cong L$, and define

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

Lattice formulation of Stanley's conjecture. For all finite semilattices $L$, the following inequalities hold:

$$\operatorname{spdim}_1 L\leq\operatorname{pdim}_1 L,$$ $$\operatorname{spdim}_2 L\leq\operatorname{pdim}_2 L,$$ $$\operatorname{spdim}_1 L\leq\operatorname{spdim}_2 L-1.$$

This reformulates the Stanley conjecture for ideals and quotients, together with the conjectured inequality relating their Stanley depths, as a statement about finite semilattices. The source gives no resolution status, so the formulation remains open here.