Lagrangian Surplusection Phenomena

Suppose you have a family of Lagrangian submanifolds $L_t$ and an auxiliary Lagrangian $K$. Suppose that $K$ intersects some of the $L_t$ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of $K$? Or will any Lagrangian isotopic to $K$ surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.