Toën's conjecture on schematic homotopy types and Tannakian gerbes

Let $k$ be a field and let $F\in LAff_k^{\sim,ffqc}$ be a schematic homotopy type. Let $i(F)$ be its image among stacks over $Hk$, and let $LParf(i(F))$ be the associated $k$-tensor Segal category. Toën's schematic Tannakian conjecture. The category $LParf(i(F))$ is $k$-Tannakian, and $FIB_k(LParf(i(F)))$ is a $k$-Tannakian gerbe. There is a natural $P$-equivalence

$$i(F)\longrightarrow FIB_k(LParf(i(F))),$$

which is universal among morphisms from $i(F)$ to $k$-Tannakian Segal gerbes. If $k$ has characteristic $0$, this morphism is an equivalence; consequently $i(F)$ is the dual gerbe, and $F\mapsto i(F)$ induces an equivalence between schematic homotopy types and $k$-Tannakian Segal gerbes. This conjecture relates schematization to the Tannakian formalism and gives the asserted stronger identification in characteristic zero.