The period characterization conjecture for three-element subtraction games

Let $S(s_1,s_2,s_3)$ be a subtraction game with $s_1<s_2<s_3$, and suppose its nim sequence $n_0n_1n_2n_3n_4\ldots$ is eventually periodic with period $p$. In Case I, assume $s_3=s_1+s_2$ and define $j$ by

$$0\leq j<2s_1,\qquad s_2-s_1\equiv j\pmod{2s_1}.$$

In Case II, assume $s_3\ne s_1+s_2$.

Period characterization conjecture. In Case I,

$$p=\begin{cases} s_2+s_3-j&\text{if }0\leq j<s_1,\\ \dfrac{s_1(s_2+s_3+j-2s_1)}{\operatorname{gcd}(s_1,2s_1-j)}&\text{if }s_1\leq j<2s_1. \end{cases}$$

In Case II, $p$ has one of seven potential values: it is a divisor of at least one of the numbers $s_i+s_j$ for $1\leq i<j\leq3$, and is exactly the greatest common divisor of all such terms. Equivalently,

$$p=\operatorname{gcd}_{(i,j)\in\mathcal{G}}(s_i+s_j),$$

where $\mathcal{G}$ is the set of pairs $(i,j)$ such that $s_i+s_j$ is a multiple of $p$.

The conjecture seeks a complete characterization of the eventual periods of nim sequences for subtraction games whose subtraction set has size three. The paper presents this as a general formula extending known results for particular families and notes that the periods remain largely mysterious; no resolution is supplied in the source.