The directed Bunkbed conjecture for the $D_3^T$ model

Let $G=(V,E)$ be a finite graph and let $T\subseteq V(G)$. In the model $D_3^T(G)$, every edge is independently assigned one of its two orientations with equal probability. A walk may switch between following and opposing the assigned directions only at vertices in $T$, and it may not use an edge in both directions. Let $u_\rightarrow$ denote departure from $u$ following an outgoing edge; at the endpoint, $\mathrel{_{\rightarrow}v}$ and $\mathrel{_{\leftarrow}v}$ indicate arrival following and opposing the edge direction, respectively.

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

$$P(u_\rightarrow\longrightarrow\mathrel{_{\rightarrow}v})\ge P(u_\rightarrow\longrightarrow\mathrel{_{\leftarrow}v}).$$

The paper introduces this as the directed analogue of the two-color bunkbed conjecture. It notes that the $D_3^T$ and $E_3^T$ models are not equivalent, and leaves the directed inequality open.