Katthän's lattice gluing conjecture for Stanley projective dimension

Let $L_2\subseteq L_1$ be finite atomistic lattices. Let $\operatorname{spdim}_\mathrm{Q} L$ denote the Stanley projective dimension of a lattice for the quotient case, and let $L_1\# L_2$ denote the lattice operation used in the paper.

Lattice gluing conjecture. If

$$\operatorname{spdim}_\mathrm{Q} L_2<\operatorname{spdim}_\mathrm{Q} L_1,$$

then

$$\operatorname{spdim}_\mathrm{Q}(L_1\# L_2)=\operatorname{spdim}_\mathrm{Q} L_1.$$

Equivalently, the lower bound established earlier in the paper is always attained. The conjecture concerns the behavior of Stanley projective dimension under the lattice operation $L_1\# L_2$; the supplied text does not provide evidence of a resolution.