Period conjecture for three-parameter Subtraction Nim

Let $G(S)$ be the normal-play Subtraction Nim game with move set $S=\{a,b,c\}$, where $0<a<b<c$ and $c\ne a+b$. Its Nim value function is considered as a function of the heap or string length.

Period conjecture. The Nim value function is periodic with a period equal to one of $a+b$, $b+c$, or $c+a$.

This is the same period claim attributed earlier in the paper to Ward, so it is merged with that conjecture. The supplied text gives no proof or resolution in general.