Unique escape-weighted measure conjecture for infinite self-avoiding walk

Let $\mu$ be the connective constant, let $\operatorname{SAW}_k$ be the set of self-avoiding paths of length $k$, and write $\eta\text{ escapes }\zeta$ for the event that the infinite path $\eta$ avoids the finite path $\zeta$ after the relevant initial segment. Unique escape-weighted measure conjecture. There exists a unique probability measure $P$ on infinite paths such that, for every $k$ and every $\zeta\in\operatorname{SAW}_k$,

$$P(\eta[0,k]=\zeta)=\mu^{-k}P(\eta\text{ escapes }\zeta).$$

The paper presents this as a slight alteration of the symmetric domain Markov property conjecture. Its resolution is not given in the supplied text.