The stronger density hypothesis for the Riemann zeta function

Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+i\gamma$ of the Riemann zeta function with $\beta\geq\sigma$ and $|\gamma|\leq T$, counted with multiplicity. For every $\varepsilon>0$, let $\delta=\delta(\varepsilon)\in(0,1)$.

Stronger density hypothesis. For every $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)\in(0,1)$ such that, for every $\nu\in[0,1/2-\varepsilon)$ and $T\geq1$,

$$N(1-\nu,T)\ll_{\varepsilon}T^{(2-\delta)\nu}.$$

This would improve the density hypothesis $N(\sigma,T)\ll_{\varepsilon}T^{2(1-\sigma)+\varepsilon}$ in the range relevant to applications, and would yield stronger large-value estimates for Dirichlet polynomials through the results of the paper. Its status is not resolved here.