Ath–Sobel conjecture on sparse uniformly most reliable graphs

Let $\mathcal C_{n,m}$ be the class of connected simple graphs on $n$ vertices and $m$ edges, and define its corank by $c=m-n+1$. A uniformly most reliable graph (UMRG) is a graph in this class whose reliability is at least that of every other member for every edge-failure probability $\rho\in[0,1]$. Ath–Sobel conjecture. If $\mathcal C_{n,m}$ is nonempty, $c\in\{5,6,7,8\}$, and $n\geq 2c-2$, then $\mathcal C_{n,m}$ contains at least one UMRG. The conjecture extends the known characterization for corank at most $4$; the supplied text gives no resolution of the stated cases.