The Shapiro-class formula for the nth prime

Let $p_n$ denote the $n$th prime, and let $S(n)$ denote the Shapiro-class function. Let $B=\gamma/\log 2$, where $\gamma$ is the Euler–Mascheroni constant. Then Shapiro-class formula for $p_n$.

$$p_n\sim\frac{1}{1+b}S(n),$$

for a constant $b$ satisfying

$$b\approx\frac{e^B}{10}\approx0.22996\dots.$$

This conjecture is motivated by numerical comparisons showing that $S(n)$ approximates the $n$th prime closely but appears to be systematically too large. Its status is unresolved in the supplied text.