The homogeneous magnetic-field direction-reversal probability conjecture

Let $a_1(x,t,u)$ denote the first component of the Feynman checker amplitude at position $x$ and time $t$ in the external field $u$, and define $u(x+1/2,t+1/2)=-1$ if both $x$ and $t$ are even and $u(x+1/2,t+1/2)=+1$ otherwise. Direction-reversal probability conjecture. The limits satisfy

$$\lim_{t\to\infty}\sum_{x\in\mathbb{Z}}a_1^2(x,2t+1,u)=\frac{\sqrt{3}}{6},$$

and

$$\lim_{t\to\infty}\sum_{x\in\mathbb{Z}}a_1^2(x,2t,u)=\frac{\sqrt{3}}{3}.$$

In contrast with the basic model, whose limiting direction-reversal probability exists, numerical experiments suggest two limit points for the model with this homogeneous external magnetic field; the stated limits describe those two subsequences.