Comparing zeros of distinct Dirichlet L-functions

For any $\theta>\frac13$, we show that there are constants $c_1,c_2>0$ that depend only on $\theta$ for which the following property holds. If $\chi_1,\chi_2$ are two distinct primitive Dirichlet characters modulo $q$, 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 $$\big\{s=\sigma+it\in{\mathbb C}:0<\sigma<1,~T<t<T+c_2q^\theta\log T\big\}.$$ For cubefree moduli $q$, the same result holds for any $\theta>\frac14$.