First-player conjecture for rectangular grid instances of INFLUENCE

Let $G_{n,m}$ be the rectangular grid with $n$ rows and $m$ columns, with alternated black and white vertices and a black vertex in the top-left corner. Let $Ls(G_{n,m})$ and $Rs(G_{n,m})$ denote the left and right scores of the corresponding game. Rectangular-grid first-player conjecture. For $n,m\geq 2$,

$$Ls(G_{n,m})>0>Rs(G_{n,m}).$$

In particular, the first player wins and there is no draw.

Despite the symmetry of rectangular grids, there is no BW-automorphism, and the computations reported in the source all give a first-player victory; the conjecture remains unproved in general.