Nonexistence of Lagrangian prism manifolds or connected sums of spherical 3-manifolds in W_k (general k)

Prove or refute the existence of closed exact Lagrangian submanifolds L ⊂ W_k (the double bubble plumbing Stein manifold as above) that are diffeomorphic either to a prism 3-manifold or to a connected sum of spherical 3-manifolds, for arbitrary integers k ≥ 1, without assuming L arises as a Dehn surgery on a knot.

Background

The authors develop tools based on cyclic quasi-dilations and group-ring arguments to exclude various spherical Lagrangian types under specific hypotheses, and they prove their main theorem by ruling out configurations arising from Dehn surgeries. However, they explicitly state that their method does not establish the nonexistence of Lagrangian prism manifolds or connected sums of spherical 3-manifolds in W_k in full generality.

This leaves open whether such Lagrangians can exist in W_k when no Dehn surgery assumption is made, which would further refine the classification of spherical Lagrangian submanifolds in these double bubble plumbings.