Main horizontal-period conjecture for Lengyel transfer games

Let $L(b;x_1,y_1;x_2,y_2)$ be the vector subtraction game with move set

$$\{(0,-b),(-x_1,y_1),(-x_2,y_2)\},$$

where $b,x_1,x_2\in\mathbb{N}^+$ and $y_1,y_2\in\mathbb{N}$. Let $g(b,x_1,0,x_2,y_2)$ denote the horizontal period of $L(b;x_1,0;x_2,y_2)$. Assume $0<y_2<2b$, $\gcd\{x_1,x_2\}=1$, and $\gcd\{b,y_2\}=1$, and assume that none of Corollary or Propositions (i), (iii),,, or applies.

Main horizontal-period conjecture. Under these assumptions,

$$g(b,x_1,0,x_2,y_2)=\frac{2b(x_1+x_2)}{\gcd\{2b,y_2\}}.$$

The claim is based on computer-aided calculations for parameters at most $20$. The paper identifies exceptional sporadic cases and states that determining when they do not occur is its main obstacle to proving this conjecture.