Finiteness conjecture for minimal symplectic fillings of planar contact manifolds

A planar contact $3$-manifold is a contact $3$-manifold supported by an open book whose pages have genus zero. A minimal symplectic filling is a symplectic filling containing no symplectic exceptional spheres, and two fillings are deformation equivalent when they lie in the same deformation class of symplectic fillings.

Planar filling finiteness conjecture. Every planar contact $3$-manifold has at most finitely many distinct deformation classes of minimal symplectic fillings.

The conjecture is motivated by the greater rigidity expected when a spinal open book is an ordinary planar supporting open book: the co-cores can then be completed to holomorphic planes with index zero and unobstructed deformation theory. The supplied text gives no resolution status, so the conjecture remains open here.