Maximum Wiener index conjecture for directed grids

Let $G_{m,n}=P_m\Box P_n$ be the Cartesian product of paths on $m,n\geq 2$ vertices. Let ${D}_{m,n}$ be the orientation of $G_{m,n}$ with all $P_m$-layers oriented up except the last $P_m$-layer, which is oriented down, and all $P_n$-layers oriented to the left except the first $P_n$-layer, which is oriented to the right. Write $W_{\rm \max}(G_{m,n})$ for the maximum Wiener index over all orientations of $G_{m,n}$, and $W({D}_{m,n})$ for the Wiener index of ${D}_{m,n}$. Maximum Wiener index conjecture. For every $m,n\geq 2$, we have

$$W_{\rm \max}(G_{m,n})=W({D}_{m,n}).$$

The conjecture asserts that the natural generalization of the orientation known to maximize the Wiener index of the ladder graph $L_n=P_2\Box P_n$ also maximizes the Wiener index for every directed grid. The ladder case was proved by Kraner Šumenjak et al.; the general case is presented as a conjecture.