Breit–Wigner series convergence conjecture for super-exponentially decreasing potentials

Let $V$ be a super-exponentially decreasing potential, meaning that for every $N\in\mathbf{N}$, $|V(x)|\lesssim_N e^{-N|x|}$. Its Breit–Wigner series is

$$B(V)=-\sum_{\lambda\in\operatorname{Res}V\setminus\{0\}}\frac{\operatorname{Im}\lambda}{|\lambda|^2}.$$

Breit–Wigner convergence conjecture. The series $B(V)$ converges if and only if $V$ is compactly supported.

For compactly supported potentials the series converges, while the conjecture predicts divergence for every super-exponentially decreasing potential that is not compactly supported. This would provide the expected cancellation of infinities in the Breit–Wigner formula as the support becomes unbounded.