The critical bias ratio conjecture for vertex-transitive graphs

Let $\Lambda$ be a connected vertex-transitive graph. Write $p_{\operatorname{c}}(\Lambda)$ for its bond percolation threshold, and let $\rho(\Lambda)$ denote the critical bias ratio of the Maker–Breaker percolation game on $\Lambda$.

Critical bias ratio conjecture. If $\Lambda$ is a connected vertex-transitive graph, then

$$\rho(\Lambda)=\frac{1}{p_{\operatorname{c}}(\Lambda)}-1.$$

The formula extends the heuristic for $\mathbb{Z}^2$: under random-like play, Maker's claimed edges should resemble bond percolation with parameter $1/(1+\rho)$, so the game threshold should coincide with the percolation threshold. Its validity for general connected vertex-transitive graphs remains open.