The nimber conjecture for Sprouts starting positions

Let $p$ be the number of spots in the starting position of a game of Sprouts, and let its nimber be the associated Sprague–Grundy value. Nimber conjecture. The nimber for the starting position with $p$ spots is $0$ if $p$ is $0$, $1$ or $2$ modulo $6$, and $1$ if $p$ is $3$, $4$ or $5$ modulo $6$. This conjecture is false: the starting position with $p=\mathord{}$ has nimber $0$, whereas the starting position with $p=\mathord{}$ has nimber $2$.