Nienhuis's conformal invariance conjecture for self-avoiding walks

Let $D$ be a domain with boundary points $A$ and $B$, and let $P_{D,A,B}^{SAW}$ denote a scaling limit of measures on self-avoiding curves from $A$ to $B$ in $D$, assigning weight proportional to $\mu^{-n(\omega)}$ to a walk $\omega$, where $n(\omega)$ is its number of steps and $\mu$ is the connective constant. Nienhuis's conjecture. The scaling limits $P_{D,A,B}^{SAW}$ exist and are conformally invariant in the sense of Proposition~. This conjecture formalizes the predicted conformal invariance of self-avoiding walks, whose scaling limit is expected to have fractal dimension $4/3$; neither existence nor the stated conformal invariance is established in the source.