Schramm–Loewner evolution scaling-limit conjecture for self-avoiding walks

Let $\Omega\ne\mathbb C$ be a simply connected domain with distinct boundary points $a$ and $b$. For $\delta>0$, let $\Omega_\delta$ be the discrete approximation in $\delta\mathbb H$, let $a_\delta$ and $b_\delta$ be the corresponding closest boundary vertices, and let $\gamma_\delta$ be sampled with probability proportional to $x^{\ell(\gamma)}$ among self-avoiding trajectories from $a_\delta$ to $b_\delta$ in $\Omega_\delta$.

Self-avoiding-walk SLE conjecture. For $x=x_c$, the law of $\gamma_\delta$ in $(\Omega_\delta,a_\delta,b_\delta)$ converges as $\delta\to0$ to chordal Schramm–Loewner Evolution with parameter $\kappa=8/3$ in $\Omega$ from $a$ to $b$.

Conformal invariance of this scaling limit would imply the predicted critical exponents $\gamma=43/32$ and $\nu=3/4$, but convergence is not proved in the source.