Quadratic characters with positive partial sums

Let ${\mathcal{L}}^{+}$ be the set of all primes $p$ for which the sums of $\left(\frac{n}{p}\right)$ over the interval $[1,N]$ are non-negative for all $N$. We prove that the estimate $$\mathrm{\mid }{\mathcal{L}}^{+}\cap [1,x]\mathrm{\mid }\ll \frac{x}{\ln x(\ln \ln x{)}^{c-o(1)}}$$ holds for $c\approx 0.0368$