Toën's full faithfulness conjecture for commutative ring spectra

Let $Sp^{\Sigma}$ be the category of symmetric spectra with its positive model structure, and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Write $L(Comm(Sp^{\Sigma}))$ for its simplicial localization and let $\underline{TensSeCat}$ denote the $2$-Segal category of tensor Segal categories. A commutative ring spectrum $A$ is sent to the tensor Segal category $L(A\text{-}Mod)$, and a morphism $A\to B$ is sent to the base-change functor $-\wedge_A^{\mathbb{L}}B$. Toën's full faithfulness conjecture. The morphism of $2$-Segal categories

$$\begin{array}{ccc} L(Comm(Sp^{\Sigma})) & \longrightarrow & \underline{TensSeCat} \\ A & \mapsto & L(A\text{-}Mod) \\ (A\rightarrow B) & \mapsto & -\wedge^{\mathbb{L}}_{A}B \end{array}$$

is fully faithful. This asserts that commutative ring spectra can be recovered from their tensor Segal categories of modules; the source presents it as the first fundamental conjecture and notes that the analogous statement for tensor triangulated homotopy categories is expected to fail.