Even-parameter Nim-sum conjecture for three-dimensional chocolate-bar games

Let $a,m\bb Z_{ 0}$, let $k=2^{a+2}m+2^{a+1}$, and define

$$f(x,z)=\left\lfloor\frac{x+z}{k}\right\rfloor.$$

Assume $x,z\leq(2^{2a+2}-2^{a+1})m+2^{2a+1}-1$. A position of the chocolate bar $CB(f,x,y,z)$ is a $\mathcal{P}$-position if and only if

$$x\oplus y\oplus z=0.$$

Even-parameter Nim-sum conjecture. The stated condition characterizes exactly the $\mathcal{P}$-positions in this restricted even-$k$ family. The authors found this claim through Mathematica calculations and had not proved it; determining the necessary and sufficient condition for the broader even-$k$ question remains difficult.