The non-integer optimality conjecture for duo-equidistant vectors

Assume $v<nx$, set $m=\left\lfloor v/x\right\rfloor$, $r=v-mx$, and $y=x-r$, and let $\Lambda(v)$, $f$, and $\Gamma(m,y,r)$ be as defined in the paper.

Non-integer optimality conjecture. The set

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

should be part of the set of optimal solutions of the minimization problem

$$\min\{f(\boldsymbol{v}):\boldsymbol{v}\in\Lambda(v)\}.$$

The statement is proved in the paper only in the case $\Delta_y=\Delta_z$; the remainder of the cases is explicitly left open.