Holomorphic-observable convergence conjecture for self-avoiding walks

Let $\Omega\subsetneq\mathbb{C}$ be a simply connected domain, let $z\in\Omega$, and let $a,b$ be distinct boundary points, with the boundary smooth near $b$. For $\delta>0$, let $F_\delta$ be the holomorphic observable in the discrete domain $(\Omega_\delta,a_\delta,b_\delta)$ approximating $(\Omega,a,b)$, and let $z_\delta$ be the closest point of $\Omega_\delta$ to $z$. Holomorphic-observable convergence conjecture. If $\Phi$ is a conformal map from $\Omega$ to the upper half-plane with $\Phi(a)=\infty$ and $\Phi(b)=0$, then

$$\lim_{\delta\rightarrow 0}\frac{F_\delta(a_\delta,z_\delta)}{F_\delta(a_\delta,b_\delta)}=\left(\frac{\Phi'(z)}{\Phi'(b)}\right)^{5/8}.$$

Convergence of the observable would provide an analytic route to identifying the scaling limit of self-avoiding walk; the discrete Morera argument explains why a continuous limit would be holomorphic, but the asserted convergence remains unproved here.