Multiplier reciprocal conjecture for Banach spaces of holomorphic functions

Let $\Omega$ be a domain, let $B(\Omega)$ be a Banach space of holomorphic functions on $\Omega$, and let $\mathcal{M}(B(\Omega))$ denote its algebra of multipliers. Multiplier reciprocal conjecture. If $\psi\in\mathcal{M}(B(\Omega))$ 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 weaker conjecture is motivated by the known corresponding result for multipliers of the weighted Bergman space $A^2_\alpha$. The general Banach-space case remains open in the source.