The limiting probability conjecture for the modified Feynman checker

Let $b_1(x,t)$ denote the first component of the modified Feynman checker amplitude at position $x$ and time $t$. Modified Feynman checker limiting-probability conjecture. The total probability satisfies

$$\lim_{t\to\infty}\sum_{x\in\mathbb{Z}}b_1^2(x,t)=\frac{\sqrt{3}}{6}.$$

This asserts existence of a time-independent limiting direction-reversal probability for the modified lattice model. The supplied excerpt contains the proof of recurrence relations used in the surrounding argument, but gives no resolution of this limiting statement.