The characterization of P-positions for triangular chocolate bar games when k is 1 modulo 4

Let $k$ be a positive integer, and let $(x,y,z)$ denote a position in the triangular chocolate bar game, with $x,y,z\in Z_{\geq 0}$. A position is a P-position if the next player has no winning strategy, and write $\oplus$ for bitwise exclusive OR. P-position characterization. When $k=4m+1$ for some $m\in Z_{\geq 0}$, $(x,y,z)$ is a P-position if and only if

$$(x-1)\oplus y\oplus (z-1)=0.$$

The statement is supported in the paper by computational evidence from CGSuite for the case $k=5$; its status is not established by the supplied text.