The truncated asymptotic bounds conjecture for the prime-counting function

Let $\pi(x)$ denote the number of primes at most $x$. Let $\underline{li}(x)$ and $\overline{li}(x)$ denote the lower and upper truncated asymptotic approximation functions defined in the paper.

Truncated asymptotic bounds conjecture. For every real $x\ge e$,

$$\underline{li}(x)\le \pi(x)\le \overline{li}(x).$$

The source proves related finite-range inequalities and presents this global inequality as a conjecture. It further states that if either this conjecture or the $li_0$-$li_1$ bounds conjecture holds, then the Riemann Hypothesis follows.