Complete score conjecture for two-row grids

Let $G_{2,m}$ be the rectangular grid with two rows and $m$ columns, and let $Ls(G_{2,m})$ and $Rs(G_{2,m})$ denote the corresponding left and right scores. Two-row grid score conjecture. For $m\geq 2$,

$$Ls(G_{2,m})=-Rs(G_{2,m})= \begin{cases} 6 & \text{if } m=3,\\ 4 & \text{if } m\equiv 1 \pmod 2 \text{ and } m\geq 5 \text{ or } m=2,\\ 2 & \text{if } m\equiv 0 \pmod 2 \text{ and } m\geq 4. \end{cases}$$

The preceding theorem and lemma establish part of this pattern, and the source notes that it remains only to prove $Ls(G_{2,m})\leq 2$ for even $m$; the conjecture was checked computationally up to $m=30$.