The LQG–SLE mating-map conjecture

Let $\kappa'\in(4,\infty)$ and $\theta\in[0,2\pi)$. Consider a $\sqrt{16/\kappa'}$-LQG on a bounded domain decorated by curves $\bm\eta$ and $\bm\eta'$ with parameters $\kappa'$ and $16/\kappa'$, respectively, and angle $\theta$. Define $\bm\psi_{\kappa',\theta}$ by

$$\bm\eta(t)=\bm\eta'(\bm\psi_{\kappa',\theta}(t)),\qquad t\in[0,1].$$

Let $\varphi_{{\bm{\mathscr Z}}_{\kappa',\theta}}$ be the function constructed from the solution flow of the generalized LQG SDE for different starting times.

LQG–SLE mating-map conjecture. Almost surely,

$$\bm\psi_{\kappa',\theta}(t)=\varphi_{{\bm{\mathscr Z}}_{\kappa',\theta}}(t),\qquad t\in[0,1].$$

This would identify the curve-interaction time change in the LQG–SLE construction with the flow generated by the conjectural SDE, extending the explicitly known special case $\kappa'=12$ and $\theta=\pi/2$.