Counterexample conjecture to the ROAD property for compact knapsack instances

For each integer $m\geq 2$, consider the integer program $(\mathrm{CE})_m$ with variables $x\in\{0,1\}^{2m}$ and its linear relaxation. Let $x_m^\circ$ denote an optimal solution of that linear relaxation, and define

$$\mathbf{X}:=x_m^\circ {x_m^\circ}^{\top}+\operatorname{Diag}\left(x_m^\circ-(x_m^\circ)^2\right).$$

The ROAD property means that this matrix is feasible for the naive semidefinite relaxation. Counterexample conjecture. For all $m\geq 2$, there is an optimal solution $x_m^\circ$ of the linear relaxation for which there exist $i,j\in\{1,\dots,2m\}$ with $j-i>2$ such that

$$\left\lfloor\frac{j-i-1}{2}\right\rfloor (x_m^\circ)_i(x_m^\circ)_j>\sum_{k=i+1}^{j-1}(x_m^\circ)_k.$$

In particular, $\mathbf{X}$ is not a solution of the naive semidefinite relaxation, and $(\mathrm{CE})_m$ does not have the ROAD property. The supplied status evidence disproves this conjecture, so the asserted universal failure has a counterexample; the exact refutation is not specified in the provided material.