Generalized non-commutative degeneration conjecture

Let $k$ be a field of characteristic zero, and let ${\mathcal B}$ and ${\mathcal C}$ be small DG categories over $k$. Let $K_n({\mathcal A})=K_n(\operatorname{Perf}({\mathcal A}))$, let $HH_{\bullet}$ denote Hochschild homology, let $HC^-_{\bullet}$ denote negative cyclic homology, and let $\delta:HH_n({\mathcal C})\to HC^-_{n+1}({\mathcal C})$ be the boundary map. Using the Chern character $ch$ and the induced tensor-product identification, define

$$\varphi_n:K_n({\mathcal B}\otimes{\mathcal C})\stackrel{ch}{\longrightarrow}(HH_{\bullet}({\mathcal B})\otimes HH_{\bullet}({\mathcal C}))_n\stackrel{\operatorname{id}\otimes\delta}{\longrightarrow}(HH_{\bullet}({\mathcal B})\otimes HC^-_{\bullet}({\mathcal C}))_{n+1}.$$

Generalized non-commutative degeneration conjecture. For every pair of small DG categories ${\mathcal B}$ and ${\mathcal C}$ over $k$, one has $\varphi_n=0$ for $n\leq 0$. This conjecture generalizes the Kontsevich–Soibelman degeneration conjecture and implies it. Its status is not resolved in the supplied text.