The nimber pattern conjecture for Sprouts on orientable surfaces

Consider a Sprouts starting position with $n$ spots on an orientable compact surface, and let its nimber be the least excluded nonnegative integer among the nimbers of the positions reachable in one move. Orientable-surface nimber conjecture. The nimber is $0$ when $n$ is congruent to $0$, $1$, or $2$ modulo $6$, and is $1$ otherwise. The conjecture generalizes the earlier winning-player pattern and is supported in the paper by computations for starting positions with up to $14$ spots and orientable genus up to $9$; it remains unproved in the source.