The Dirichlet-series Hardy-space duality conjecture

Let $p\leq2$, let $\mathcal{H}^p$ denote the Hardy space of Dirichlet series, and let $\varphi_{1/p}$ be the Dirichlet series defined in the source by equation. Dirichlet-series Hardy-space duality conjecture. The Dirichlet series $\varphi_{1/p}$ defines a bounded linear functional on $\mathcal{H}^p$ if and only if $p\leq1$. This is the Dirichlet-series analogue of the preceding duality result for Hardy spaces on the unit disc. The conjecture is stated as an open problem for the range $p\leq2$.