Montgomery's $\ell^q$ conjecture for Dirichlet polynomials

Let $M_{Dir}$ be the matrix associated with Dirichlet polynomials $D(t)=\sum_{n=N}^{2N}b_n e^{i t\log n}$, with $|b_n|\leq1$, and suppose $N\leq T\leq N^{O(1)}$. Montgomery's $\ell^q$ conjecture. For every $q\geq2$,

$$\|M_{Dir}\|_{\infty\rightarrow q}\lessapprox N+N^{1/2}T^{1/q}.$$

Equivalently,

$$\sum_{t=1}^T|D(t)|^q\lessapprox N^q+N^{q/2}T.$$

The source describes this as equivalent to a sharp $\ell^q$ estimate underlying a weaker form of the density hypothesis; no resolution is given.