Categorical characterization of spherical adjunctions by relative S-constructions

Let $\operatorname{Sph}$ be the $\infty$-category formed by spherical adjunctions, and let ${\mathcal{S}}t_{\Lambda_\infty}^{\operatorname{adm},\,2\text{-Segal}}$ be the $\infty$-category formed by stable $\infty$-categories and admissible $2$-Segal paracyclic stable $\infty$-categories. Categorical characterization conjecture. The relative $S$-construction should induce an equivalence

$$\operatorname{Sph}\longrightarrow {\mathcal{S}}t_{\Lambda_\infty}^{\operatorname{adm},\,2\text{-Segal}}.$$

The paper proves that a spherical adjunction produces an admissible $2$-Segal paracyclic stable $\infty$-category; the conjecture asks for the converse and categorical equivalence.