The conjectured RH inequality for primes congruent to 3 modulo 8

Let $p$ and $q$ be primes with $p\equiv q\equiv 3 \pmod 8$, satisfying $0<2a<p<q$, and let $\chi_q(p)=\left(\frac{p}{q}\right)$ be the Legendre symbol modulo $q$. Let

$$denote the equation referenced in the source. **RH inequality conjecture.** The relation$$

holds for all such primes $p$ and $q$. The source gives no resolution; the relation is intended as a criterion connected with the Riemann Hypothesis.