Small quadratic non-residue conjecture

Let $p$ be a prime. A quadratic non-residue modulo $p$ is an integer whose Legendre symbol modulo $p$ is $-1$.

Small quadratic non-residue conjecture. There exists a prime $q<2\sqrt{p}$ such that

$$q\equiv 3\pmod{4}$$

and $q$ is a quadratic non-residue modulo $p$.

A positive answer would remove the dependence on the Generalized Riemann Hypothesis in the construction of graphs realizing odd-order finite abelian groups with perfect pairings. The source gives no resolution evidence.