Band induction conjecture for tripod Nim bands

Let $C_c(a,b)$ denote the quantity associated with the tripod Nim array parameterized by $c$, and say that the numbers from $0$ through $m$ form a band when the corresponding band condition holds. Fix $c \in \mathbb{N}$. Band induction conjecture. If there exist $N,k$ such that, for all $a,b>N$,

$$|a-b| \leq 4k-1 \iff C_c(a,b) \leq 8k-2,$$

then there exists $N'$ such that, for all $a,b>N'$,

$$|a-b| \leq 4k+3 \iff C_c(a,b) \leq 8k+6.$$

This conjecture asserts an inductive extension of a band from radius $4k$ to radius $4k+4$; the surrounding discussion relates it to the stability of periodic orbits of $D(8,4n)$, but no proof or resolution is supplied here.