The Edge-Wilson 25% speed-up conjecture for transitive graphs

Let $G$ be a transitive graph. Wilson's algorithm is said to take asymptotically half as much time to construct the first branch when initialized with an edge rather than a single vertex if the corresponding first-branch construction time is asymptotically half that for a single-vertex initial condition.

Edge-Wilson speed-up conjecture. If Wilson's algorithm has this property on $G$, then the expected time taken by Edge-Wilson$(G)$ is asymptotically $25\%$ less than the expected time taken by Wilson$(G,\varnothing)$, Wilson's algorithm on $G$.

The conjecture formalizes the heuristic that Edge-Wilson saves half the time spent constructing the first branch while requiring the same time as Wilson's algorithm to complete the tree. The stated examples motivating it include complete graphs, complete balanced bipartite graphs, and high-dimensional hypercubes; no resolution is given here.