Natural $S^{1}$-equivariant lift of the trace for derived categorical sheaves

Let $X$ be a scheme or algebraic stack, let $T\in Dg_{parf}(X)$ be a perfect derived categorical sheaf, and let $p:\mathrm{L}X\longrightarrow X$ be the natural morphism. The pullback $p^{*}(T)$ has an autoequivalence $u$, whose trace is a perfect complex

$$Tr(u)\in \underline{End}_{Dg_{parf}(\mathrm{L}X)}(1)=D_{parf}(\mathrm{L}X).$$

Natural lift conjecture. The complex $Tr(u)$ has a natural lift

$$Tr^{S^{1}}(u)\in D^{S^{1}}_{parf}(\mathrm{L}X),$$

where $D^{S^{1}}_{parf}(\mathrm{L}X)$ is the $S^{1}$-equivariant perfect derived category of $\mathrm{L}X$. The conjecture requires a natural lift, not merely the existence of some lift.

This lift is intended to provide the $S^{1}$-equivariant refinement needed in the construction of the Chern character for derived categorical sheaves. The source notes that it is difficult to characterize the required lift by specific properties; the conjecture is resolved by the subsequent application to rigid symmetric monoidal $(1,\infty)$-categories.