The multiple-of-a-power-of-two conjecture for the Sprague–Grundy function

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$, and assume $x_1\leq x_2$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. The multiple-of-a-power-of-two conjecture. If, for some $j\in\mathbb{Z}_{\geq}$, either

$$x_0+x_2\geq j2^k\geq x_2$$

or

$$x_2-x_0\leq j2^k\leq x_2,$$

then $\mathcal{G}(x)=u(x)$. The claim describes the experimentally observed attainment of the upper bound when $x_2$ lies within distance $x_0$ of a multiple of $2^k$, under the displayed restriction on $x_1$; the restriction is essential, and the conjecture is open.