Dual characterization question for heaviness via the counit morphism

Determine whether, under the hypotheses of Theorem 5.1 on F and the idempotent e, the implication holds that if a compact subset A ⊂ T^*M is ζ_e-heavy, then the counit morphism ε_{A,F} : ι_A^!F → F satisfies c(e ε_{A,F}; ι_A^!F, F) ≠ +∞ in h𝒯_∞(T^*M).

Background

Theorem 5.1 provides several equivalent criteria for ζ_e-heaviness, primarily via the left adjoint ι_A^* and the unit morphism. A natural dual formulation would use the right adjoint ι_A^! and the counit morphism.

The author establishes many equivalences but notes that the dual implication from heaviness to a nontrivial spectral condition on the counit is not known; moreover, an analogous dual to the key technical proposition fails, leaving the overall dual statement unresolved.