Weak Mertens conjecture

Let $\mu(n)$ be the Möbius function and define the Mertens function by

$$M(x)=\sum_{n\le x}\mu(n).$$

Weak Mertens conjecture.

$$\int_1^T\left(\frac{M(x)}{x}\right)^2\,dx\ll\log T.$$

The source states that this conjecture implies the Riemann Hypothesis, simplicity of all zeta zeros, and a bound involving $\zeta'(\rho)$. It is described as generally believed but is not resolved in the supplied text.