Presentability of limit- and filtered-colimit-closed full subcategories

Let $\mathcal{C}$ be a presentable $\infty$-category and let $\mathcal{C}_0\subseteq\mathcal{C}$ be a full subcategory. Suppose that $\mathcal{C}_0$ is closed under small limits and $\kappa$-filtered colimits for some regular cardinal $\kappa$. Presentability conjecture. Then $\mathcal{C}_0$ is presentable. For ordinary presentable categories this was proved, while the corresponding assertion for presentable $\infty$-categories is expected to hold although the classical proof does not generalize straightforwardly.