For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ where $\sigma \leq \beta$, $\lvert \gamma\rvert \leq T$. We know that the upper bounds of $\sum_{\chi}N(\sigma, T, \chi)$, are called as the zero-density estimates for $L(s, \chi)$. Well, several results exist in literature for such estimates, both asymptotic and explicit. For example, Chen showed that for $q\geq 3$, $\frac{1}{2}\leq \sigma <1$, $T \geq \max(\frac{10^5}{q}, 10^4 \log q)$, $\sum_{\chi} N(\sigma, T, \chi) \leq \Bigl( \frac{25039}{\log(qT)} + 5700\Bigr)(q^3 T^4)^{1-\sigma} (\log(qT))^{6\sigma}$, where $\chi$ runs through all non-principal characters $\mod q$, and subsequently other such explicit results also exist.

My question is whether or not strict inequalities between $\frac{1}{2}$ and $\sigma$ or between $\sigma$ and $\beta$ would make any difference to the results achieved for $\sum_{\chi} N(\sigma, T, \chi)$. Is it that under the GRH, it makes a theoretical difference but that might not show up in the estimation results used to obtain the explicit results as above?