Infinity of the τ-invariant under dual-cone boundary vanishing for Maslov-zero tori

Show that if L ⊂ M is a Maslov 0 Lagrangian torus and the boundary of the dual cone C^*(L,M) satisfies ∂(C^*(L,M)) = {0}, then the τ-invariant associated to L equals infinity; that is, prove τ(L) = ∞.

Background

The τ-invariant, defined as the valuation of the leading deformation differential d_def in the special deformation retraction model, measures the size of ambient deformation of SC^*_M(D) relative to the intrinsic theory. Earlier, Theorem \ref{tmTauConstant} shows spectral sequence degeneration when the dual cone C^*(D,M) projects trivially under the boundary map, in exact contexts.

The conjecture proposes removing exactness assumptions, positing that for Maslov 0 Lagrangian tori L with ∂(C^*(L,M)) = {0}, one should have τ(L) = ∞. This would imply that ambient symplectic cohomology is undeformed relative to intrinsic models around such tori, significantly strengthening the local-to-global reconstruction framework in SYZ mirror symmetry.

A proof would extend ambient–intrinsic degeneracy criteria beyond exact or aspherical settings and reinforce the role of flux-based cone conditions in controlling deformations.