Heaviness-to-counit implication for characterization of heaviness

Ascertain whether the implication from heaviness to the counit condition holds: under the assumptions of Theorem 5 (cohomologically constructible F and bounded q_ℝπ(ΜS(F)∩{τ>0})), determine whether, for any compact subset A ⊂ T^*M and idempotent e: F → F with c(e; F, F) ≠ +∞, the ζ_e-heaviness of A implies that 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

Section 5 establishes an equivalence between ζe-heaviness and certain unit morphism conditions involving the left adjoint ι_A^*. The dual statements involving the right adjoint ι_A^! and the counit ε{A,F} are natural counterparts.

While the paper proves the equivalences using the unit morphism (conditions (i), (ii), (iii)), it leaves open whether heaviness also implies the corresponding counit-based criterion (condition (v)). A counterexample shows the dual of a key proposition fails, but the full implication (i) ⇒ (v) might still hold under the stated hypotheses.