Duchêne et al.'s ultimate periodicity conjecture for segment scores

A segment of $n$ vertices starting with a black vertex is denoted by $S_n$, and a segment starting with a white vertex by $S_{-n}$. Let $Ls(S_n)$ and $Rs(S_n)$ denote the left and right scores of the segment game on $S_n$. Duchêne et al.'s conjecture. The sequences $Ls(S_n)$ and $Rs(S_n)$ are ultimately periodic.

Exact score values are known for some values of $n$, but the majority of the scores remain unknown; computations of the first 80 values suggest this periodicity.