Conjecture on the asymptotic formula for the nth prime

For each positive integer $n>2$, define

$$\lambda_n=\lceil 2(\log n!+1)\rceil,\qquad \mu_n=\lfloor \log \log n!\rfloor.$$

Here $p_n$ denotes the $n$th prime number. Asymptotic formula for the nth prime. There exist some $\epsilon\in[0,1]$ and $\delta\in[-2,2]$ such that

$$p_n=\left\lceil \lambda_n-\mu_n^{2+\epsilon}+\delta\right\rceil;$$

in particular, for all $n>2$,

$$p_n\in\left(\lambda_n-\mu_n^3-2,\,\lambda_n-\mu_n^2+2\right).$$

This conjecture proposes a precise approximation to the $n$th prime in terms of factorial logarithms. The supplied text does not provide evidence that the conjecture has been proved or disproved.