Coefficient-growth conjecture for the Cauchy transform

Let $\nu$ be a measure for which the Cauchy transform has Taylor expansion

$$\mathcal{C}_\nu(z)=\sum_{n\geq 0}\nu_nz^n.$$

Let $\{M_n\}_{n\geq 0}$ be a logarithmically convex sequence satisfying

$$\sum_{n\geq 0}\frac{\bigl(\log(1/M_n)\bigr)^2}{1+n^2}=\infty.$$

Coefficient-growth conjecture. The conclusion of the uncertainty theorem referred to as \thref{UncertThmRSD} can be reached if merely

$$\sup_{n\geq 0}\frac{|\nu_n|}{M_n}<\infty.$$

This proposes that the theorem's conclusion remains valid under the displayed coefficient bound and divergence condition, rather than the stronger hypothesis used in the theorem; the source does not state whether it is resolved.