Symmetric domain Markov property conjecture for infinite self-avoiding walk

Let $d\geq1$, and let $\operatorname{SAW}_\infty$ be the space of infinite self-avoiding walks. For a walk $\eta$, write $\eta[i,\infty)$ for its tail and $\eta[0,k]=\zeta'$ for its initial segment. Symmetric domain Markov property conjecture. For all $d\geq1$, there exists a unique probability measure $P$ on $\operatorname{SAW}_\infty$ such that

$$\mathbb{P}(\eta(1)=e)=\frac{1}{2d}$$

for every neighbour $e$ of the origin, and such that, for all events $E$,

$$P(\eta[k,\infty)\in E\mid\eta[0,k]=\zeta')=P(\eta[0,\infty)\in E\mid\eta[1,\infty)\cap\zeta'=\varnothing,\,\eta(0)=\zeta(k)).$$

In lower dimensions, existence of one-sided infinite self-avoiding walk has not been proved. The conjecture provides an alternative characterization, while identifying it with the weak limit of finite self-avoiding walks would require proving that subsequential limits satisfy the symmetric domain Markov property.