A counterexample to the Jordan-Hölder property for polarizable semiorthogonal decompositions

We show that the Jordan-Hölder property fails for polarizable semiorthogonal decompositions – those where every factor admits a Bridgeland stability condition. Counterexamples exist among Fukaya categories of surfaces and bounded derived categories of smooth projective varieties. Furthermore, we give an example of a smooth and proper pre-triangulated dg category with positive rank Grothendieck group which does not admit a stability condition.