Boesch's conjecture on uniformly most reliable graphs

Let $\mathcal{C}_{n,m}$ be the set of all connected simple graphs on $n$ vertices and $m$ edges. For $G\in\mathcal{C}_{n,m}$, let $N_i^{(1)}(G)$ denote the number of spanning subgraphs of $G$ with $i$ edges and at most one connected component. A graph $G$ is a 0-element in $\mathcal{C}_{n,m}$ if

$$N_i^{(1)}(G)\geq N_i^{(1)}(H)$$

for every $i\in\{0,1,\ldots,m\}$ and every $H\in\mathcal{C}_{n,m}$. A graph is uniformly most reliable if it is a $1$-uniformly most reliable graph, meaning that its reliability is at least that of every graph in $\mathcal{C}_{n,m}$ for every edge-retention probability $p\in[0,1]$.

Boesch's conjecture. Each uniformly most reliable graph in $\mathcal{C}_{n,m}$ is a $0$-element in $\mathcal{C}_{n,m}$.

Every $0$-element is uniformly most reliable, since its coefficientwise inequalities imply the corresponding reliability inequalities. The converse, proposed by Boesch in 1986, remains unresolved.