Positive linear lower bound for the specific matrix triple

Prove that there exists a constant $C>0$ such that $|L_0A+L_1A+L_2A|\geq C|A|$ for every finite set $A\subset\mathbb{Z}^3$, where $L_0,L_1,L_2$ are the explicitly defined non-pre-commuting matrices.

Background

The proposed eightfold bound would imply a positive linear lower bound, but the authors report that even this substantially weaker assertion is unresolved. Establishing such a bound would provide a first quantitative result for the example and a possible entry point toward the general non-pre-commuting problem.

The matrices in question are stated to be irreducible and coprime, so the difficulty arises specifically from the failure of the pre-commutation hypothesis rather than from the known degeneracies excluded by those conditions.