Comparing zeros of distinct Dirichlet $L$-functions

For any $\theta > \frac13$, we show that there are constants $c_1,c_2 > 0$ depending only on $\theta$ for which the following property holds. If $\chi_1$, $\chi_2$ are two distinct primitive Dirichlet characters $\mod q$, and $T\ge c_1q^\theta$, then $L(s,\chi_1)$ and $L(s,\chi_2)$ do not have the same zeros in the region $$\mathcal{R}:=\big\{s=\sigma+it\in\mathbb{C}:0 < \sigma < 1,T < t < T+c_2q^\theta\log T\big\}.$$ For cubefree moduli $q$, the same result holds for any $\theta > \frac14$.