Bernardara–Marcolli–Tabuada's noncommutative nilpotence conjecture

Let $\mathcal{A}$ be a smooth and proper dg category, and let $K_0(\mathcal{A})$ be the Grothendieck group of its category of compact objects. Let $\sim_{\otimes_{\tiny\text{nil}}}$ denote the equivalence relation induced by tensor-nilpotence and let $\sim_{\otimes_{\tiny\text{num}}}$ denote the equivalence relation induced by numerical triviality. Bernardara–Marcolli–Tabuada's noncommutative nilpotence conjecture. One has

$$K_0(\mathcal{A})/_{\sim_{\otimes_{\tiny\text{nil}}}}=K_0(\mathcal{A})/_{\sim_{\otimes_{\tiny\text{num}}}}.$$

This is the noncommutative analogue of Voevodsky's nilpotence conjecture, replacing algebraic cycles by the Grothendieck group of a smooth proper dg category. The paper invokes this conjecture in its study of noncommutative K3 surfaces; the supplied text does not establish a general resolution.