The eventual upper-bound conjecture for Tetris Nim

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. The eventual upper-bound conjecture. If either $x_0>1$ or $x_0=1$ and $x_1$ is odd, and if $x_2$ is sufficiently large, then $\mathcal{G}(x)=u(x)$. The conjecture formalizes the observed eventual regularity of the Sprague–Grundy function for large third coordinates, but “sufficiently large” is not quantified and the claim is unproved.