CLE carpet and stable-carpet identification conjecture

Set $\alpha \defeq 1/2+4/\kappa$. Let $(\mathcal{D},\Phi,0,\infty)$ be a $\gamma$-Liouville quantum gravity disk, and let $\Gamma$ be an independent non-nested $\operatorname{CLE}_\kappa$ in $\mathcal{D}$. Write $X$ for the carpet of $\Gamma$, and let $\mu_{\Phi,X}$ and $D_{\Phi,X}$ denote the induced measure and metric on $X$. CLE carpet and stable-carpet identification conjecture. Conditional on $\mu_{\Phi,X}(X)=1$, the metric measure space $(X,D_{\Phi,X},\mu_{\Phi,X})$ is conditionally a disk variant of the $\alpha$-stable carpet. This conjectural identification links CLE carpet metrics coupled to LQG disks with stable carpets arising as scaling limits of Boltzmann planar maps.