Non-displaceability conjecture for fibers of symplectic cluster 4-manifolds

Let $X$ be a symplectic cluster $4$-manifold equipped with a cluster presentation $\{\pi,L_1,\dots,L_n\}$. A fiber means a fiber of $\pi$ over a point of the base. Non-displaceability conjecture. For any $1\leq i\leq n$, no fiber of $\pi$ over a point in $\pi(L_i)$ can be displaced from $L_i$ by a Hamiltonian isotopy.

This conjecture is motivated by mirror symmetry, the cited results on cluster embeddings and Lagrangian tails, and the developing theory of the local Fukaya category. It is presented as a special case of the expectation that the support of a tropical Lagrangian should equal the defining tropical variety; the general expectation has only initial steps known.