Distance-exponent monotonicity conjecture for CLE carpet metrics

For $\kappa \in (8/3,4)$, let $\theta(\kappa)$ denote the distance exponent associated with the canonical geodesic metric on the $\operatorname{CLE}_\kappa$ carpet. Distance-exponent monotonicity conjecture. The assignment

$$(8/3,4) \mapsto \mathbb{R} \colon \kappa \mapsto \theta(\kappa)$$

is continuous and strictly increasing, and $\theta(\kappa) \to 1$ as $\kappa \to (8/3)^+$. The exact exponent is not known; the paper notes a Monte Carlo prediction but does not establish this conjecture.