  Hooley conjectured that $G(x;q) \ll x\log q$, as soon as $q\to +\infty$,
where $G(x;q)$ represents the variance of primes $p \leq x$ in arithmetic
progressions modulo $q$, weighted by $\log p$. In this paper, we study
$G_\eta(x;q)$, a function similar to $G(x;q)$, but including the weighting
factor $\eta\left(\frac{p}{x}\right)$, which has a dampening effect on the
values of $G_\eta$. Our study is motivated by the disproof of Hooley's
conjecture by Fiorilli and Martin in the range $q \asymp \log \log x$. Even
though this weighting factor dampens the values, we still prove that an
estimation of the form $G_\eta(x;q) \ll x\log q$ is false in the same range.
