LI-conditioned extremal-order conjecture for the Chebyshev psi function

Let $\psi(x)=\sum_{p^k\leq x}\log p^k$ be the weighted prime-counting function. Assume the Riemann hypothesis, and let the nontrivial zeros of the zeta function be written as $\rho_n=1/2+i\gamma_n$. The LI-conditioned extremal-order conjecture. If the imaginary parts $\gamma_n$ satisfy the stated LI conjecture, then

$$\liminf_{x\to\infty}\frac{\psi(x)-x}{\sqrt{x}(\log\log x)^2}=-\frac{1}{\pi},\qquad \limsup_{x\to\infty}\frac{\psi(x)-x}{\sqrt{x}(\log\log x)^2}=\frac{1}{\pi}.$$

This refines the usual conditional bounds for $\psi(x)-x$ and depends on both the Riemann hypothesis and linear independence of the zero ordinates; the supplied text does not establish it.