The noncommutative cohomology conjecture for dg-categories

Let $\mathbb{TR}$ be a candidate (co)homology theory on dg-categories over $\mathbf{F}_{p}$, with $\mathsf{dg}\mathsf{Cat}_{\mathbf{F}_{p}}$ the category of dg-categories over $\mathbf{F}_{p}$ and $\mathsf{Sp}$ the category of spectra. For a Morita equivalence $\mathsf{C}\to\mathsf{D}$, require that $\mathbb{TR}(\mathsf{C})\to\mathbb{TR}(\mathsf{D})$ be a stable equivalence, and let $K(-)$ denote algebraic $K$-theory. If $X$ is a sufficiently nice $\mathbf{F}_{p}$-scheme and $l\neq p$ is prime, write $\mathsf{Perf}(X)$ for its dg-category of perfect complexes, $F$ for a Frobenius transformation, and $(-)^{hF}$ for homotopy fixed points. The noncommutative cohomology conjecture. There exists a (co)homology theory $\mathbb{TR}$ satisfying Morita invariance, being a localizing invariant, admitting a natural transformation $K(-)\to\mathbb{TR}(-)$, giving $\pi_{\ast}\mathbb{TR}(\mathsf{C})$ a natural cochain-complex structure, and such that for every such $X$ and $l\neq p$ its completed hypercohomology recovers $l$-adic cohomology, it admits Frobenius and recovers the Lefschetz–Grothendieck $l$-adic trace formula, and the induced trace map $K(\mathsf{Perf}(X))\to\mathbb{TR}(\mathsf{Perf}(X))^{hF}$ satisfies

$$\pi_iK(\mathsf{Perf}(X))\otimes\mathbf{Z}_l\cong\pi_i(\mathbb{TR}(\mathsf{Perf}(X))^{hF})\otimes\mathbf{Z}_l$$

for $i\geq 0$. The conjecture proposes a noncommutative analogue of $l$-adic cohomology and the Quillen–Lichtenbaum approximation, with the trace map providing the connective approximation after $l$-completion.