Hochschild functor conjecture for enriched categories and bimodules

Let $(\mathscr{V},\otimes,I)$ be a presentably symmetric monoidal $(\infty,1)$-category. Let $\mathscr{M}\mathrm{od}_{\mathscr{V}}$ be the $(\infty,2)$-category whose objects are $\mathscr{V}$-enriched categories and whose morphisms are bimodules. There is a Hochschild construction $\mathrm{biHH}_\infty(\mathscr{M}\mathrm{od}_{\mathscr{V}})$. Hochschild functor conjecture. There exists a functor of $(\infty,1)$-categories

$$\mathrm{HH}_{\mathscr{V}}:\mathrm{biHH}_\infty(\mathscr{M}\mathrm{od}_{\mathscr{V}})\to\mathscr{V}.$$

Moreover, if $\mathscr{V}$ is closed symmetric monoidal, then $\mathrm{HH}_{\mathscr{V}}$ is a $\mathscr{V}$-enriched functor of $\mathscr{V}$-enriched $(\infty,1)$-categories. This would supply the Hochschild shadow in the important case of enriched categories and bimodules; the source does not state whether the claim has been proved or remains open.