The componentwise characterization 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 $\Delta_y$ and $\Delta_z$ be the spacing parameters used in the paper, and write $\boldsymbol{v}=\boldsymbol{v}_y+\boldsymbol{v}_r$.

Non-integer characterization conjecture. A vector $\boldsymbol{v}\in\Lambda(v)$ is optimal if and only if it has this decomposition and satisfies, for the relevant indices $l$,

$$\sum_{i=0}^{\Delta_y-1}v_{y,l+i}\leq y,\qquad \sum_{i=0}^{\Delta_y}v_{y,l+i}\geq y,$$

with $v_{y,l}=0$ when $l\leq\Delta_y-1$ or $l\geq n-(\Delta_y-1)$, together with

$$\sum_{i=0}^{\Delta_z-1}v_{z,l+i}\leq y,\qquad \sum_{i=0}^{\Delta_z}v_{z,l+i}\geq y,$$

with $v_{z,l}=0$ when $l\leq\Delta_z-1$ or $l\geq n-(\Delta_z-1)$. The claim is presented as a conjectural description of all optimal solutions; no resolution is supplied.