Non-atomicity conjecture for distances between typical CLE carpet points

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 probability measure $\mu_X$, and let $D_X$ be the canonical metric on $X$. Distance non-atomicity conjecture. For every deterministic $a \ge 0$,

$$D_X(x,y) \ne a$$

almost surely. This conjecture is related to uniqueness of the geodesic between typical points and is noted to be substantially less immediate for CLE metrics than for the LQG metric.