The convex-hull characterization conjecture in the integer case

Assume $v<nx$ and that $m\in\mathbb{N}$ satisfies $v=mx$. Let $\Delta_x$ be as defined in the paper, and let $\Lambda(v)$ and $\Gamma(m,x,0)$ denote the feasible set and structured-vector set, respectively. Suppose $\boldsymbol{v}\in\Lambda(v)$ satisfies the preceding theorem's characterization of optimal solutions.

Integer-case convex-hull conjecture. Every such vector should satisfy

$$\boldsymbol{v}\in\operatorname{Conv}(\Gamma(m,x,0)).$$

The preceding theorem gives an if-and-only-if characterization of optimality when $r=0$; this conjecture asserts that those conditions describe precisely the convex hull of the structured vectors. No proof or resolution is given.