Toën's structure conjecture for Tannakian Segal categories

Let $k$ be a field and let $T$ be a $k$-Tannakian Segal category. Let $\mathcal{H}(T)$ denote a $k$-Tannakian category and let $Ind(\mathcal{H}(T))$ be its ind-category with its induced $k$-tensor structure. Toën's structure conjecture. There exist a $k$-Tannakian category $\mathcal{H}(T)$ and a non-negatively graded $E_{\infty}$-algebra $A$ in the $k$-tensor category $Ind(\mathcal{H}(T))$ such that $T$ is equivalent to the Segal category of $E_{\infty}$-modules over $A$ in $Ind(\mathcal{H}(T))$. This would provide a structure theorem for $k$-Tannakian Segal categories over a field; the source presents it as conjectural and motivates it using the heart and the adjunction between the Segal categories of Ind-objects.