Conjecture on nim-values for the divide-and-residue game

Consider the divide-and-residue game, and let ${\mathcal{SG}}(n)$ denote the nim-value of a heap of size $n$. Divide-and-residue conjecture. Every nim-value occurs for at least one heap size and for at most finitely many heap sizes. Moreover,

$$\frac{{\mathcal{SG}}(n)}{n}\longrightarrow 0$$

as $n\longrightarrow\infty$. The statement combines the finite-occurrence claim with an asymptotic bound on the nim-values; the source supplies no resolution of these assertions.