Lawler–Schramm–Werner scaling-limit conjecture for self-avoiding walks

Let $\Omega$ be a simply connected domain in $\mathbb{C}$ with boundary points $a$ and $b$, and let $(\Omega_\delta,a_\delta,b_\delta)$ be discrete approximations. At the critical fugacity $z=z_c$, let $\omega_\delta$ be the self-avoiding walk from $a_\delta$ to $b_\delta$ in $\Omega_\delta$, sampled with Boltzmann weight proportional to $z^{\ell(\omega)}$. Lawler–Schramm–Werner conjecture. For $z=z_c$, the random curve $\omega_\delta$ converges to $\mathrm{SLE}_{8/3}$ from $a$ to $b$ in the domain $\Omega$. This would establish the predicted conformally invariant scaling limit of critical self-avoiding walk; existence and conformal invariance of that scaling limit remain open, although the identification of any such limit with $\mathrm{SLE}_{8/3}$ is supported by the work of Lawler, Schramm, and Werner.