Uniform minimization conjecture for light-traffic coefficients

Consider a redundancy system with $N$ servers and edge-selection probabilities $P=(p_{\{i,j\}})_{i,j}$, where $i,j\in\{1,\ldots,N\}$ and $i\ne j$. For each $q\geq 0$, let $\alpha_q(P)$ denote the coefficient governing the light-traffic probability of having at least $q$ jobs. Let $P'=(p'_{\{i,j\}})_{i,j}$ be the uniform policy, with

$$p'_{\{i,j\}}=\frac{1}{\binom{N}{2}}.$$

Uniform minimization conjecture. For every $q\geq 0$, the coefficient under the uniform policy equals the minimum over all edge-selection probability vectors:

$$\alpha_q^*\coloneqq\alpha_q(P')=\min\left\{\alpha_q(P)\mid P=(p_{\{i,j\}})_{i,j}\right\}.$$

This optimization conjecture would imply the uniform sampling dominance conjecture in the light-traffic regime through the stated comparison of tail probabilities. The coefficients are continuous functions on the compact set of edge-selection probabilities, so a minimum exists; what remains open is proving that the uniform policy attains it for every $q$.