Prime-restricted asymptotic formula for the maximum packing number

Let $\mathbb{P}(x)$ be the set of prime numbers not exceeding $x$, and let $M_{\mathbb{P}(n)}(n)$ denote the corresponding prime-restricted maximum packing number. Prime-restricted maximum-packing conjecture.

$$M_{\mathbb{P}(n)}(n)\sim \frac{1}{6}\cdot \frac{n^{3}}{\ln n}.$$

The source presents this as a prime-restricted problem closely related to the preceding conjecture; its lower bound follows from the proof of an earlier theorem, but the asymptotic equality is open.