Two walkers crossing a 4 by 4 grid meeting

On the square grid with corners $(0,0)$ and $(4,4)$, Alan starts at $(0,0)$ and Barbara starts at $(4,4)$, and both move once per minute at the same time. Alan steps one unit right or one unit up with equal probability, except that when he is on an edge of the square where one of those moves would leave it, he takes the other. Barbara does the same with steps left or down.

What is the probability that they are at the same point at some moment before they reach each other's starting corner?

$\tfrac{70}{256}=\tfrac{35}{128}$. Alan's coordinate sum goes up by one each minute and Barbara's goes down by one, so they can only meet at minute $4$ on the line $x+y=4$, before any boundary forcing matters. Each reaches point $k$ on that line with probability $\binom4k/16$, so the answer is $\sum_k\binom4k^2/256=\binom84/256$.