I am asking this question on MO as earlier it was asked on MSE (link below). I even put a bounty on it and waited but no one answered. So I am posting it here as I have no option and I am unable to think about it. (It was asked on MSE on 13 Feb, 2020) .

https://math.stackexchange.com/questions/3543021/unable-to-think-how-to-prove-a-result-in-a-research-paper-of-wadim-zudilin-and-t

Question is ->I am studying research paper: A note on odd zeta values and I am unable to think how to deduce a result which the authors don't prove. This result has to be proved assuming the prime number theorem and it's on Page 12 of the paper :

Prove that $\lim_ {n\to\infty} \frac{\log(\Phi_n) } {n} =\int_0^{1} \rho_0 (t) d(\psi(t) + 1/t) $, where $\psi(t) $ = $\frac {\Gamma'(t) } {\Gamma(t) } $.

where $\Phi(n)$ and $\rho(n)$ are described in this image:

Can someone please tell how to prove this result ?

I shall be really thankful.