Erdős's eventual-time conjecture for quadratic character sums

Let $obreak\footcomment$ be omitted. For an odd \prime $p$, write

$$S_\ell(p):=\sum_{n\leq \ell}\left(\frac{n}{p}\right),$$

where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol modulo $p$. For $\varepsilon>0$, let $F_\varepsilon(p)$ be the smallest integer such that

$$S_\ell(p)<\varepsilon\ell\qquad\text{for every }\ell\geq F_\varepsilon(p).$$

Erdős's eventual-time conjecture. There exists a constant $C_\varepsilon>0$ such that

$$\sum_{p\leq x}F_\varepsilon(p)\sim C_\varepsilon\,\frac{x}{\log x}.$$

Erdős posed this as a question about the eventual time at which a quadratic character sum remains below a linear barrier. The conjecture was proved by Elliott.