The duality conjecture for the Dirichlet series $\varphi_{1/p}$ on $\mathcal{H}^p$

Let $0<p\leq2$, and let $\varphi_{1/p}$ be the Dirichlet series defined in the paper by the referenced formula. The space $\mathcal{H}^p$ denotes the corresponding Hardy space of Dirichlet series. Duality conjecture. The Dirichlet series $\varphi_{1/p}$ defines a bounded linear functional on $\mathcal{H}^p$ if and only if $0<p\leq1$. This conjecture is proposed as the Dirichlet-series analogue of the preceding duality result for Hardy spaces on the disk; its resolution is not specified in the source.