The Newman–Shapiro problem and spectral synthesis in Fock space

In 1966 D. Newman and H. Shapiro posed the following problem. Let $G$ be a function from Fock space $\mathcal{F}$ and such that $e^{zw}G\in\mathcal{F}$ for any $w\in\mathbb{C}$. Is it true that
$$ \operatorname{Span}\{FG: FG \in \mathcal{F}\}=\operatorname{Span}\{e^{wz}G:w\in\mathbb{C}\}? $$
Recently the author (joint with A. Borichev) has constructed a counterexample to this conjecture. On the other hand, we are able to show that if $G$ satisfies some regularity conditions, then conjecture holds. These results are clоsеly connected to some spectral synthesis problems in Fock space.