The structured-vector optimality conjecture for the continuous queueing problem

Let $x,w,n,m,r$ be the parameters of the paper, set $y=y(x,w)=x-r$, and define $\tau_1=\tau_u(n,m)$ and $\tau_2=\tau_u(n,m+1)$. For any vectors

$$\boldsymbol{v}_r\in\Gamma_{\tau_1}(r,(m+1)r),\qquad \boldsymbol{v}_y\in\Gamma_{\tau_2}(y,my),$$

let

$$\boldsymbol{v}^*(\boldsymbol{v}_y,\boldsymbol{v}_r)=\boldsymbol{v}_y+\boldsymbol{v}_r.$$

Structured-vector optimality conjecture. The vector $\boldsymbol{v}^*$ is an optimal solution of the paper's continuous minimization problem.

The conjecture unifies the integer and non-integer constructions and was verified in many numerical examples. The source explicitly leaves its proof as an open problem; it also notes that some parameter regimes are trivial or already covered by earlier theorems.