Decomposability threshold conjecture for Legendrian doubles

Let $\lambda$ be a Legendrian link such that $\mathcal{M}_1(\lambda)$ has a cluster structure. Let $L_1$ and $L_2$ be exact Lagrangian fillings of $\lambda$, and let $d_\mu(L_1,L_2)$ be their mutation distance. Let $\Lambda(L_1,L_2)$ be their Legendrian double. Decomposability-threshold conjecture. There exists an $N$ such that

$$d_\mu(L_1,L_2)>N$$

implies that $\Lambda(L_1,L_2)$ is not decomposable. This generalizes the paper's result for trivalent graph fillings of $\lambda(2,n)$, where sufficiently large mutation distance obstructs decomposability; the general statement remains open.