Constrained random model conjecture for the prime-counting function

Let ${\operatorname{RM}}_c$ denote the constrained ${\operatorname{RM}}$ described in the source, and let $\pi_{\operatorname{RM}_c}(x)$ be its prime-counting function. Let $\operatorname{li}(x)=\int_2^x dt/\log t$ be the logarithmic integral. Constrained RM asymptotic conjecture. The expected value satisfies

$$\mathbf{E}\left[\pi_{\operatorname{RM}_c}(x)\right]\sim \operatorname{li}(x).$$

This conjecture asserts that imposing the product constraint on the random model restores the prime number theorem asymptotic, unlike the unconstrained model whose expected counting function is asymptotic to $2\mathrm{e}^{-\gamma}\operatorname{li}(x)$. The source provides numerical evidence but no proof or resolution.