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  • $\begingroup$ I was hesitant to commit to your Claim 2 because I was thinking about asymptotics for $f(n)$, and I am unfamiliar with the underlying methods. But a saddle-point argument for the upper bound, while too lossy for an asymptotic for $f(n)$, should be plenty to get an asymptotic for $\log f(n)$. Very nice! $\endgroup$ Commented Dec 2, 2021 at 20:46