Almost-sure uniqueness conjecture for typical CLE carpet geodesics

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant measure $\mu_X$, normalized to be a probability measure, and let $D_X$ be the canonical metric on $X$. Typical-geodesic uniqueness conjecture. Almost surely, there is a unique $D_X$-geodesic connecting $x$ and $y$. This is an open question about geodesic uniqueness in the random CLE carpet metric.