The enriched presheaf conjecture for presentable monoidal infinity-categories

Let $\mathcal{V}$ be a presentable, closed monoidal $\infty$-category. Write $\operatorname{Cat}^{\mathcal{V}}$ for the $\infty$-category of $\mathcal{V}$-enriched categories and $\operatorname{RMod}_{\mathcal{V}}(\operatorname{Pr}^L)_\star$ for the $\infty$-category of presentable right $\mathcal{V}$-modules with some distinguished objects. Enriched presheaf conjecture. The $\infty$-category

$$\operatorname{Cat}^{\mathcal{V}}$$

embeds as a full subcategory of

$$\operatorname{RMod}_{\mathcal{V}}(\operatorname{Pr}^L)_\star.$$

This conjecture would reduce enriched higher category theory to presentable $\infty$-categories and make enriched $\infty$-categories more usable in areas such as secondary algebraic K-theory and algebraic K-theory of analytic rings. The source presents it as conjectured by experts and gives no resolution.