Comparing zeros of distinct Dirichlet L-functions

For any $\theta >\frac{1}{3}$, 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 $T\ge c_1{q}^{\theta }$, then $L(s,{\chi }_1)$ and $L(s,{\chi }_2)$ do not have the same zeros in the region $$\{s=\sigma +it\in \mathbb{C}:0<\sigma <1,T<t<T+c_2{q}^{\theta }\log T\}\mathrm{.}$$ For cubefree moduli $q$, the same result holds for any $\theta >\frac{1}{4}$.