Toën's characterization conjecture for Tannakian Segal categories

Let $A$ be a commutative ring spectrum, let $T$ be an $A$-tensor Segal category, and let $FIB_A(T)$ be its stack of fiber functors. For an $A$-Tannakian Segal category, the source considers the natural morphism $T\to LParf_A(FIB_A(T))$. Write $Ind(T)$ for the Segal category of Ind-objects and let $\omega:T\to LParf(A')$ be a fiber functor after an $sffqc$ covering $A\to A'$. Toën's characterization conjecture. (i) For any $A$-Tannakian Segal category $T$, the natural morphism

$$T\longrightarrow LParf_A(FIB_A(T))$$

is an equivalence of $A$-tensor Segal categories. (ii) A rigid $A$-tensor Segal category is Tannakian precisely when the structural morphism induces $LParf(A)_{(1,1)}\simeq T_{(1,1)}$ and there is an $sffqc$ covering $A\to A'$ and an $A$-tensor morphism $\omega:T\to LParf(A')$ whose Ind-extension is conservative and satisfies the stated orthogonality characterization of $Ind(T)_{\geq 0}$ by $Ind(T)_{<0}$. This is the main conjectural duality criterion in the source; the source does not establish the equivalence in general.