Gonek's conjecture on zeros of the Hurwitz zeta function

Let $a$ be a real number with $0<a<1$, and let $\zeta(s,a)$ denote the Hurwitz zeta function. For $T>0$, consider the segment from $1/2$ to $1/2+iT$ in the complex plane, and count zeros of $\zeta(s,a)$ there with multiplicity. Gonek's conjecture. If $a$ is rational and $a\ne 1/2$, then the number of zeros of $\zeta(s,a)$ on the segment $(1/2,1/2+iT)$ is $\ll T$. This conjecture extends the known upper bounds for the specified rational parameters and remains open for general rational $a\ne 1/2$.