Existence of a proper singular variety with Hochschild homology nonzero in only finitely many degrees

Ascertain whether there exists a proper singular variety X such that the Hochschild homology groups HH_n(Perf(X)) are nonzero for only finitely many integers n.

Background

Motivated by the lack of a geometric counterexample to the DG analogue of Han’s conjecture, the authors pose a weaker existence question in the algebro-geometric setting. They highlight that for projective nodal or cuspidal curves C over \mathbb{C}, HH_n(C) \neq 0 for all n \ge -1, so these do not provide the desired example.

This question seeks a proper singular variety whose category of perfect complexes has Hochschild homology vanishing in all but finitely many degrees, a property analogous to the finiteness condition appearing in Han’s conjecture.