Cycle-complement conjecture for dense uniformly most reliable graphs

Let $C_m$ denote a cycle of length $m$, let $\cup$ denote disjoint union, and let an overline denote graph complementation. A uniformly most reliable graph (UMRG) is a graph with the highest all-terminal reliability among graphs having the same order and size, for every edge-failure probability. Write $n=3k+r$ with $0\leq r<3$.

Cycle-complement conjecture. For any $n\geq 5$ and $e=n(n-3)/2$, the graph

$$\overline{(k-1)C_3\cup C_{3+r}}$$

is a UMRG.

The conjecture was verified in the paper for $n\leq 11$, with strong computational evidence for $n=12,13,14$. Its general case remains open in the supplied text.