Toën's schematization and Tannakian duality conjecture

Let $k$ be a field and let $(X\otimes k)^{sch}\in LAff_k^{\sim,ffqc}$ be the schematic homotopy type associated with a connected finite CW complex $X$, and let $LParf(X,k)$ be its tensor Segal category of perfect complexes. Let $i$ embed stacks over $k$ into stacks over $Hk$. Toën's schematization and Tannakian duality conjecture. There is a natural morphism

$$i(X\otimes k)^{sch}\longrightarrow FIB_k(LParf(X,k))$$

which is a $P$-equivalence; consequently

$$LParf(X,k)\longrightarrow LParf(i(X\otimes k)^{sch})$$

is an equivalence. If $k$ has characteristic $0$, the first morphism is an equivalence. This identifies the schematization with the Tannakian dual in characteristic zero, while in general only the asserted $P$-equivalence and categorical consequence are claimed.