The i-Mark conjecture on positions with Sprague–Grundy value two

Consider the impartial game $\operatorname{i\text{-}Mark}(\{s\},\{d\})$ in normal play, with relatively prime positive integers $s$ and $d$. Let $\operatorname{SG}(n)$ denote the Sprague–Grundy value of the position $n$, and define the sequences

$$a_0=2ds,\qquad a_{i+1}=d(a_i+s),$$ $$b^j_0=jd,\qquad b^j_{i+1}=d(b^j_i+s)\quad (1\leq j\leq s-1).$$

i-Mark conjecture. A necessary condition for $\operatorname{SG}(n)=2$ is that one of the following holds: $n=sd$, $n$ is a term of $(a_n)$, or $n$ is a term of one of the sequences $(b^j_n)$ for $1\leq j\leq s-1$. The remaining Sprague–Grundy values iterate in $s$-tuples of zeros and ones.

The conjecture is supported by the authors' experiments and describes the expected structure of the Sprague–Grundy sequence for general relatively prime subtraction and division parameters. No proof or resolution is supplied in the stated context.