Strictness of the weak-product inclusion for Dirichlet-series spaces

Let $\mathscr{H}^2$ be the Hilbert space of Dirichlet series with square-summable coefficients, let $\odot$ denote the weak product, and let $\partial$ be the differentiation operator on Dirichlet series. The space $\partial^{-1}(\mathscr{H}^2\odot\partial\mathscr{H}^2)$ is the corresponding skew weak product space.

Strict weak-product inclusion conjecture. The inclusion between the standard weak product and its skew counterpart is strict:

$$\mathscr{H}^2\odot\mathscr{H}^2 \subsetneq \partial^{-1}\left(\mathscr{H}^2\odot\partial\mathscr{H}^2\right).$$

The analogous strict inclusion is proved for $\mathscr{H}^2_0$, the subspace of Dirichlet series with vanishing constant term. The proof does not extend directly to $\mathscr{H}^2$ because the relevant matrices are Hilbert--Schmidt, so the conjectured strictness for the standard space remains open.