The factorial growth of topological recursions

We show that the \(n\)-point, genus-\(g\) correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like \((2g)!\) as \(g \to \infty\), which is the expected growth rate. This provides, in particular, an upper bound for many curve counting problems in large genus and serves as a preliminary step for a resurgence analysis.