Functorial $S^{1}$-equivariant trace for rigid symmetric monoidal $\mathbb{S}$-categories

Let $T$ be a rigid symmetric monoidal $\mathbb{S}$-category. For an object $x\in T$ with an autoequivalence $u$, the trace construction gives a morphism from the loop simplicial set $LT$ to $T(1,1)$, where $LT$ is the $S^{1}$-equivariant simplicial set formed from invertible morphisms in $T(S^{1})$ and $T(1,1)$ has the trivial $S^{1}$-action. Let $\mathbb{S}\text{-}\operatorname{Cat}^{rig}$ be the category of rigid symmetric monoidal $\mathbb{S}$-categories, and let

$$L,E:\mathbb{S}\text{-}\operatorname{Cat}^{rig}\longrightarrow S^{1}\text{-}SSet$$

be the functors sending $T$ to $LT$ and to $T(1,1)$, respectively.

Functorial trace conjecture. There exists a morphism

$$Tr:L\longrightarrow E$$

in $Ho(\operatorname{Fun}(\mathbb{S}\text{-}\operatorname{Cat}^{rig},S^{1}\text{-}SSet))$ such that, for every rigid symmetric monoidal $\mathbb{S}$-category $T$, the induced morphism

$$Tr:LT\longrightarrow T(1,1)$$

is the trace map described above.

This conjecture gives a functorial $S^{1}$-equivariant refinement of the trace map, which is the categorical input for the natural lift in the derived-geometric construction. The source says that its application to rigid symmetric monoidal $(1,\infty)$-categories solves the preceding conjecture, so this conjecture is resolved.