Generalized reflectional periodicity conjecture for odd Inverse k-cross games

Let $k=2m+1$ with $m\geq1$, and consider an Inverse $k$-cross game with empty-board state $[a]$. Let $\mathcal{G}([a])$ denote its Sprague–Grundy value, and let $\oplus$ denote nim-sum. Generalized reflectional periodicity conjecture. For every such game, $\mathcal{G}([a])\leq1$, and

$$\mathcal{G}([2+3m+a])=\mathcal{G}([a])\oplus(m\bmod2),$$ $$\mathcal{G}([2+5m-a])=\mathcal{G}([a])\oplus(m\bmod2).$$

For $k=2$, eventual periodicity is known through the identification with Dawson's Kayles, whereas for even $k>2$ eventual periodicity is unknown; the asserted uniform bound and reflectional periodicity for odd constraints remain conjectural.