Montgomery's conjecture on large values of Dirichlet polynomials

For $1/2<\sigma\leq 1$ and $\tau\geq 0$, let $\mathrm{LV}(\sigma,\tau)$ denote the large-values exponent for Dirichlet polynomials in the source: it measures the exponent governing the proportion of parameters $t$ in a range of length $N^{\tau+o(1)}$ for which a polynomial of length $N$ can have size at least $N^{\sigma+o(1)}$. Montgomery's conjecture.

$$\mathrm{LV}(\sigma,\tau)=\min(2-2\sigma,\tau)$$

for all $1/2<\sigma\leq 1$ and $\tau\geq 0$. Equivalently,

$$\mathrm{LV}(\sigma,\tau)\leq 2-2\sigma$$

for all such $\sigma$ and $\tau$. The conjecture asserts that the lower bound obtained by random-sign constructions is sharp; the source gives no resolution status.