Reciprocal conjecture for Banach spaces of holomorphic functions

Let $\Omega$ be a domain and let $B(\Omega)$ be the Banach space of holomorphic functions on $\Omega$. Reciprocal conjecture. If $\psi\in B(\Omega)$ and there exists a constant $c>0$ such that

$$|\psi(z)|\geq c$$

for all $z\in\Omega$, then $1/\psi\in B(\Omega)$. This is proposed as an analogue of Zhu's reciprocal problem for weighted Bergman spaces. The source presents it as a conjecture for a general Banach space of holomorphic functions; its resolution is not specified.