The planar loop-erased random walk neighbor probability conjecture

Let $x \in \mathbb{Z}^2$ be a neighbor of the origin, and let $p_N$ be the probability that $x$ lies on the loop-erased random walk from $o$ to $\partial G_N$. Loop-erased random walk neighbor probability conjecture. As $N \to \infty$,

$$p_N \to \frac{5}{16}.$$

This predicts an exact limiting probability for a fixed neighbor of the origin to lie on a loop-erased random walk in the planar lattice. The supplied context does not state whether the claim has been proved or remains open.