Equivalent Riemann hypothesis conjecture for the constrained random model

Let $\operatorname{RM}$ be the random model, with a realisation determined by the shifts $a$ and factors $\rho_k(n+a)$, and let $\pi_{\operatorname{RM}}(x)$ be its counting function. Let $\operatorname{Ri}(x)$ be the Riemann prime-counting function. A realisation belongs to the constrained model $\operatorname{RM}_c$ only if

$$\prod_{k=0}^{\pi(\sqrt n)}\rho_k(n+a)\leq n$$

for every relevant value of $n$. Equivalent RH conjecture. If a realisation of the RM satisfies

$$\left|\pi_{\operatorname{RM}}(x)-\operatorname{Ri}(x)\right|^2\neq O\left(x(\log x)^2\right),$$

then there exists at least one value of $n$ such that

$$\prod_{k=0}^{\pi(\sqrt n)}\rho_k(n+a)>n.$$

Thus, any realisation violating the stated error bound is not an element of $\operatorname{RM}_c$. The source presents this as an equivalent formulation related to the Riemann hypothesis, but supplies no proof or resolution.