Equivalence between ζ_MVZ-superheaviness and cohomological superheaviness

Determine whether every ζ_MVZ-superheavy compact subset A ⊂ T^*M is cohomologically superheavy, i.e., whether the morphism β μ_M : 𝕜_{M×[0,∞)} → ι_A^* 𝕜_{M×[0,∞)} ⊗ or_{M×ℝ}[n] is necessarily nonzero in h𝒯_∞(T^*M), where β is the unit of the adjunction ι_A^* ⊣ ι_A*.

Background

The paper introduces "cohomologically superheavy" as a sheaf-theoretic strengthening of superheaviness defined via the unit map to ι_A^* and the fundamental class μ_M. It proves that cohomological superheaviness implies ζ_MVZ-superheaviness.

However, it remains unclear whether the converse holds in general. The author later shows that assuming the Viterbo conjecture, ζ_MVZ-heaviness implies cohomological superheaviness, but without that conjecture the equivalence is open.