General recognition conjecture for $s \ell$-tannakian categories

Let $B\in\operatorname{Alg}_{s \ell}$, let $\mathcal{A}$ be an $s \ell$-enriched bounded category, and let $\mathcal{A}\xrightarrow{F}(B\text{-}\operatorname{Mod})_0$ be a functor. Let $L$ be the object associated to $F$ as in the preceding construction, and let $\widetilde{F}$ denote the resulting lifting.

General recognition conjecture. Under these hypotheses, if $F$ is an $s \ell$-enriched open and faithful functor, then $\widetilde{F}$ is an equivalence.

This is proposed as a generalisation of the recognition theorem previously obtained for the particular tannakian setting of relations of a topos, with comodules over a cogebroid not necessarily arising from a Hopf cogebroid. The authors present it as future work and as a conjectural direction for $s \ell$-enriched categories.