The first friendly-versus-antagonistic discrepancy outside the consecutive-ratio family

Let $S=\{s_2,s_1\}$ with $s_2<s_1<2s_2$, and suppose

$$\frac{s_1}{s_2}\neq\frac{k+1}{k}$$

for every $k\in\mathbb{N}$. Let FvF and AvA denote the symmetric friendly and antagonistic self-interest games. First-discrepancy conjecture. There exists a heap with a friendly/antagonistic discrepancy, and the first such heap is

$$x=(i+2)s_2+iks_1,$$

where $i\in\mathbb{N}$ satisfies

$$\frac{i+1}{i}>\frac{s_1}{s_2}>\frac{i+2}{k+i}.$$

This conjecture complements the claimed no-discrepancy family above and is supported by the paper's analysis; it remains unresolved in the supplied text.