The eventual periodicity conjecture for the exceptional even case

Let $x^*_0=1$ and let $x^*_1$ be even with $2^{k-1}<x^*_1<2^k$ for some $k\in\mathbb{Z}_{\geq}$. Define the defect $\Delta(x^*_0,x^*_1,x_2)=u(x^*_0,x^*_1,x_2)-\mathcal{G}(x^*_0,x^*_1,x_2)$. The eventual periodicity conjecture. As a function of $x_2$, the defect takes only even values and is periodic with period $2^k$ once $x_2$ is sufficiently large. This conjectures structured eventual behavior precisely in the even-$x^*_1$, $x^*_0=1$ case excluded by the preceding eventual upper-bound claim; no proof or threshold is given.