Grand Density Hypothesis for Selberg class $L$-functions

Let $\mathcal{L}$ belong to the subclass $\mathcal{S}'$ of the Selberg class, and let $N_\mathcal{L}(\sigma,T)$ denote the number of zeros $\rho_\mathcal{L}=\beta_\mathcal{L}+i\gamma_\mathcal{L}$ of $\mathcal{L}$ satisfying $\beta_\mathcal{L}>\sigma$ and $|\gamma_\mathcal{L}|<T$. Grand Density Hypothesis. There is some positive constant $c_\mathcal{L}$ such that, for $\sigma>1/2$,

$$N_\mathcal{L}(\sigma,T)\ll T^{1-c_\mathcal{L}(\sigma-1/2)+\varepsilon}.$$

This conjectural density estimate strengthens the zero-density assumption defining $\mathcal{S}'$ and would control zeros of Selberg class $L$-functions close to the critical line. Its status is not resolved in the supplied source.