The numerator nondivisibility conjecture for W(q,x)

Let $q$ be a prime with $q\equiv 3 \pmod 8$, let $x$ be rational with $0<x<1/2$, and define

$$W(q,x):=h(q)-\sum_{n\leq qx}\chi_q(n)\left(1-\frac{n}{qx}\right),$$

where $h(q)$ is the class number and $\chi_q$ is the relevant quadratic character. Numerator nondivisibility conjecture. The numerator of $W(q,x)$ is not divisible by $q$. If true, the paper states that this would imply the Riemann Hypothesis. No resolution is given.