Strict Schnirelmann-density conjecture for visibility sets

Let $S\subseteq\mathbb{Z}^k$ be a finite set, and let $V(S)$ denote the set of lattice points mutually visible from every point of $S$. Write $D(V(S))$ for its asymptotic density and $\operatorname{SD}(V(S))$ for its Schnirelmann density. Strict-density conjecture. If

$$D(V(S))>0,$$

then

$$\operatorname{SD}(V(S))<D(V(S)).$$

The conjecture concerns whether positive-density visibility sets always have strictly smaller Schnirelmann density. The paper establishes the strict inequality for several specific families, but leaves the assertion for arbitrary finite $S$ open.