Period-$2m$ conjecture for arithmetic-progression Max-Welter positions

Let $a\geq 0$, $m\geq 1$, and $k\geq 1$. Then the sequence

$$\{\mathcal{G}(a+i,a+m+i,\ldots,a+km+i)\}_{i\geq 0}$$

is ultimately periodic with period $p=2m$. This conjectures periodicity of the Sprague–Grundy values for positions whose coordinates form an arithmetic progression with common difference $m$.