The majority nonnegativity conjecture for Legendre-symbol sums

Let $p$ be an odd prime, let $0\leq\alpha\leq\frac{1}{2}$, and define

$$L(\alpha,p)=\sum_{n\leq\alpha p}\left(\frac{n}{p}\right),$$

where $\left(\frac{n}{p}\right)$ is the Legendre symbol. Majority nonnegativity conjecture. For every $0\leq\alpha\leq\frac{1}{2}$,

$$\liminf_{x\to+\infty}\frac{1}{\pi(x)}\#\left\{p\leq x:\sum_{n\leq\alpha p}\left(\frac{n}{p}\right)\geq0\right\}\geq\frac{1}{2}.$$

Thus, for each such $\alpha$, at least half of the primes asymptotically should satisfy $L(\alpha,p)\geq0$. Numerical computations for several values of $\alpha$ suggest an even larger proportion, but no proof of the stated lower-density bound is given.