The generalized Bunkbed conjecture for two-color edge percolation

Let $G=(V,E)$ be a finite graph, let $T\subseteq V(G)$, and let $\tilde G=G\times K_2$. In the model $E_3^T(\tilde G)$, every horizontal edge is assigned independently to the downstairs or upstairs layer with equal probability, and vertical edges are present exactly at vertices in $T$. Equivalently, color every edge of $G$ red or blue independently with equal probability; a walk may change color only at a vertex of $T$.

Two-color Bunkbed conjecture. For every $u,v\in V(G)$,

$$P(u_0\longleftrightarrow v_0)\ge P(u_0\longleftrightarrow v_1).$$

This is a stronger-looking reformulation of the conditioned model and is proved in the paper for some graph classes. Its validity for arbitrary graphs and arbitrary $T$ remains open.