The odd-cycle terminating-game nimber conjecture

Let $C_n$ be the cycle graph on vertices $v_1,\ldots,v_n$, where $n\geq 4$ and each $v_i$ is adjacent to $v_{i+1}$ with indices taken modulo $n$. Let $\operatorname{TER}(C_n)$ denote the terminating geodetic removing game on $C_n$, and let $\operatorname{nim}(\operatorname{TER}(C_n))$ be its nimber. Odd-cycle terminating-game conjecture. For cycle graphs with odd $n$,

$$\operatorname{nim}(\operatorname{TER}(C_n))=0.$$

The conjecture concerns the case not covered by the preceding strategy for even cycles; it has been verified computationally up to $n=21$, while the general odd-cycle case remains open.