The convex-hull inclusion conjecture for non-integer optimal vectors

Assume $v<nx$, let $m=\left\lfloor v/x\right\rfloor$, $r=v-mx$, and $y=x-r$. Let $\Gamma(m,y,r)$ be the structured-vector set defined in the paper, and suppose $\boldsymbol{v}\in\Lambda(v)$ satisfies the preceding conjectural characterization.

Non-integer convex-hull conjecture. Every vector satisfying that characterization should belong to

$$\operatorname{Conv}(\Gamma(m,y,r)).$$

This would identify the vectors described by the componentwise conditions with the convex hull of the duo-equidistant vectors. The claim is not proved or otherwise resolved in the supplied text.