The fair cake-cutting conjecture for uniformly most reliable hamiltonian graphs

Let $FCG_{n,c}$ denote the fair cake-cutting graph with parameters $n$ and $c$. A graph is uniformly most reliable for fixed order $n$ and size $m$ if its reliability polynomial is at least that of every competing graph for every edge-operational probability $p\in[0,1]$. Fair cake-cutting conjecture. If

$$n\equiv 0 \pmod{2c},$$

then $FCG_{n,c}$ is a uniformly most reliable hamiltonian graph for $m=n+c$. The case $m\geq n+4$ is described as largely open in the paper, with only some small cases known computationally; this conjecture proposes a family that is optimal under the stated divisibility condition.