The Density Hypothesis for the Riemann zeta-function and Dirichlet $L$-functions

Let $\sigma\in[1/2,1]$, let $q\geq 2$ be an integer, and let $T\geq 3$. Define the zero-counting sets

$$N(\sigma,T)=\{\rho\in\mathbb{C}:\zeta(\rho)=0,\ \operatorname{Re}(\rho)\geq\sigma,\ |\operatorname{Im}(\rho)|\leq T\},$$

and

$$N(\sigma,T,\chi)=\{\rho\in\mathbb{C}:L(\rho,\chi)=0,\ \operatorname{Re}(\rho)\geq\sigma,\ |\operatorname{Im}(\rho)|\leq T\}.$$

Density Hypothesis. The implied constants being absolute, one should have

$$N(\sigma,T)\ll T^{2(1-\sigma)}\log(T)$$

and

$$\sideset{}{^*}\sum_{\chi\ (\mathrm{mod}\ q)}N(\sigma,T;\chi)\ll(qT)^{2(1-\sigma)}\log(qT).$$

The hypothesis predicts the expected density of zeros near the critical line for the zeta-function and the family of Dirichlet $L$-functions. The source presents it as a conjecture but gives no resolution status.