Montgomery's large value conjecture for Dirichlet polynomials

Let $\sigma \geq 1/2$ and let

$$D(t)=\sum_{N < n \leq 2N}a_{n}n^{it}$$

with $|a_n|\leq 1$. Suppose that $W\subset [0,T]$ is a $1$-separated set such that $|D(t)|\geq N^\sigma$.

Montgomery's large value conjecture. One should have

$$|W|\lesssim_{\sigma,\epsilon}T^\epsilon N^{2-2\sigma}.$$

This conjecture remains beyond current techniques. It would imply the density hypothesis for the Riemann zeta function, and has significant applications including a proposed route to the Kakeya set conjecture.