Fully packed $O(2)$ map model scaling-limit conjecture

For $k\in\mathbb N$, let $(M_k,L_k)$ be sampled from the probability measure on triangulations of the disk decorated by fully packed $O(2)$ loop configurations, at the critical parameter $\beta=\beta_c$. Here $M_k$ has boundary perimeter $k$, and the loop configuration $L_k$ is on the dual map. Fully packed $O(2)$ scaling-limit conjecture. Under an appropriate scaling, as $k\to\infty$, $(M_k,L_k)$ converges in law to a critical ($\gamma=2$) LQG disk with unit boundary length together with an independent CLE$_4$. This is described as a well-known folklore conjecture and remains open.