The conformal-limit interface probability conjecture for critical site percolation

Let $\Omega_h$ be a polygon whose boundary consists of edges of a hexagonal lattice of meshsize $h$, with distinct side midpoints $a_h,b_h$ and lattice-edge midpoints $u_h,v_h$. Color each hexagon independently black or white with probability $1/2$. For each coloring, let $\gamma$ be the interface from $a_h$ to $b_h$ such that hexagons on its left are white and those on its right are black, and define $P(u_h\cdots v_h,a_h\leftrightarrow b_h)$ and $P(a_h\leftrightarrow b_h,u_h\cdots v_h)$ as the probabilities that $u_h,v_h$ lie in the same component to the left and right of $\gamma$, respectively. Interface probability conjecture. As $h\to0$ and $(\Omega_h,a_h,b_h,u_h,v_h)$ approaches the unit circle with boundary points $a,b$ and points $u,v\ne a,b$ in the Carathéodory sense, the difference of these probabilities converges to

$$\frac{2\Gamma(2/3)}{\sqrt{\pi}\Gamma(1/6)}\operatorname{Im}\left(\frac{1+\eta}{1-\eta}\,{}_2F_{1}\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\left(\frac{1+\eta}{1-\eta}\right)^2\right)\right),$$

where

$$\eta=\frac{a-u}{b-u}\cdot\frac{1-b\bar v}{1-a\bar v},$$

and ${}_2F_{1}$ is the branch on $\mathbb{C}-[1,+\infty)$ analytically continued from $D^2$, with cut values taken from the upper or lower half-plane according to whether $u,v$ lie on the counterclockwise or clockwise boundary arc $ab$. This is a precise scaling-limit prediction for an interface observable in critical percolation. The supplied text gives no resolution, so the conjectured convergence remains open.