Conjecture on intersection lattices of symplectic fillings

Let $Y$ be a contact $3$-manifold satisfying the assumptions of Corollary~, and let $X$ be a symplectic filling of $Y$. Let $J_X$ denote the intersection lattice of $X$, and let $\widetilde J_X$ denote its associated lattice after quotienting by torsion as in the corollary.

Intersection-lattice conjecture. The conclusion of Corollary~ still holds, under the same assumptions, if $X$ is allowed to range over the set of all symplectic fillings of $Y$.

The conjecture proposes that the finiteness conclusion previously obtained for the relevant intersection lattices of fillings remains valid without restricting to the class considered in the corollary. The source gives no evidence of a resolution.