Period formula conjecture for subtraction games with FES sets of size 3

Let $a$ and $b$ be positive integers such that $b>3a$ and $\gcd(a,b)=1$. Consider the nim sequence for the finite subtraction set $a,b,a+b$ and let $p$ denote its period.

Period formula conjecture. If there exists a multiple $m$ of $2a$ satisfying

$$b<m<a+b,$$

then $p=3am$. If no such $m$ exists, then there is some other integer $n$ satisfying

$$b<n<a+b$$

such that $p=3an$.

The preceding theorem establishes pure arithmetic periodicity for every subtraction set of size $3$, but the period length is not determined in general. This conjecture proposes a specific form for the period when $b>3a$, after reducing to relatively prime $a$ and $b$ using the scaling behavior of these games.