The Shapiro-class formula for the prime-counting function

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

$$G(n)=\frac{n^2}{S(n)}+\frac{(bn)^2}{S(\lfloor bn\rfloor)}.$$

Shapiro-class formula for $\pi(n)$. Then $G(n)\sim\pi(n)$ for a constant $b$, where

$$b\approx \frac{e^B}{10}\approx 0.22996\dots.$$

This conjecture refines the comparison of the prime-counting function with the Shapiro-class function and is motivated by the observed error in approximating $\pi(n)$ by $n^2/S(n)$.