The generalized Bunkbed conjecture for conditioned edge percolation

Let $G=(V,E)$ be a finite graph and let $\tilde G=G\times K_2$ be its bunkbed graph. Let $T\subseteq V(G)$ be the set of vertices whose vertical edges are present. Let $\boldsymbol p=(p_e)_{e\in E(G)}$ be a probability vector with $0\le p_e\le 1$. In the model $E_2^{\boldsymbol p,T}(\tilde G)$, each pair of horizontal edges corresponding to $e\in E(G)$ is independently present with probability $p_e$.

Generalized Bunkbed conjecture. For every $u,v\in V(G)$,

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

This extends the original conjecture by conditioning on an arbitrary set of present vertical edges and allowing edge-dependent horizontal probabilities. The paper proves the assertion when $|T|\leq 1$, while the general statement is left open.