Classification conjecture for invariant vectors in CircularNim games

A CircularNim game is denoted by $\operatorname{CN}(n,k)$, and an invariant vector is a vector associated with the set of $\mathcal{P}$-positions as defined for these games. Let $\bm{1}$ denote the all-ones vector.

Classification conjecture. The only CircularNim games that admit invariant vectors are

$$\operatorname{CN}(3k,k+1),\quad \operatorname{CN}(3k+1,k+1),\quad \operatorname{CN}(3k+2,k+1)$$

with $k\geq 1$, together with $\operatorname{CN}(n,n-1)$. More specifically, $\bm{1}$ is one of the invariant vectors.

This conjecture proposes a complete classification of CircularNim games admitting invariant vectors, based on the structural and computational evidence discussed in the paper. The cited cases have been solved, but the claimed exclusivity remains open.