Breit–Wigner divergence alternative under Froese's hypotheses

Let $V$ be a super-exponentially decreasing potential, let $\widehat V$ have completely regular growth, and let $B(V)$ be its Breit–Wigner series. Assume that $V$ is not compactly supported. Breit–Wigner divergence alternative. Either $B(V)$ diverges, or $V$ is a counterexample to Froese's conjecture.

This is presented as a weaker form of the paper's stronger Breit–Wigner convergence conjecture. It links divergence of the resonance series for noncompactly supported potentials to the possible failure of Froese's resonance-counting estimate.