The power-of-two conjecture for the Sprague–Grundy function of Tetris Nim

Let $x=(x_0,x_1,x_2)$ be a position with $x_0\in\mathbb{Z}_{\geq}$ and $x_2\geq x_1=2^k$ for some $k\in\mathbb{Z}_{\geq}$. Write $\ell(x)=x_0+(x_1\mathbin{\oplus}x_2)$ for the lower bound and $u(x)=x_0+x_1+x_2$ for the upper bound. The power-of-two conjecture. If $x_2=(2j+1)2^k+m$ with $j\in\mathbb{Z}_{\geq}$ and $0\leq m<2^k-x_0$, then $\mathcal{G}(x)=\ell(x)$; otherwise $\mathcal{G}(x)=u(x)$. Computations indicate that when the second coordinate is a power of $2$, the Sprague–Grundy function always equals one of these two bounds according to this rule.