Existence of an infinite global dimension algebra that is not of infinite + global dimension

Determine whether there exists a bound quiver algebra Λ of infinite global dimension that is not of infinite + global dimension; concretely, decide whether there exists Λ with infinite global dimension such that for every pair of vertices (y,x) with yΛx ≠ 0, the graded vector space Tor^Λ_*(k_x,{}_yk) is finite (i.e., non-infinite).

Background

In Subsection 6.4, the authors analyze the implications of a potential counterexample to the implication “infinite global dimension ⇒ infinite + global dimension.” They explain that such an example would provide a counterexample to Han’s conjecture, and they note its nonexistence is consistent with the current lack of counterexamples to Han’s conjecture.

This frames a concrete existence question: whether there is a bound quiver algebra of infinite global dimension that nonetheless fails to be of infinite + global dimension.

References

Of course we do not know of such an example since, up to date, there are no known counterexamples to Han’s conjecture.

— Happel's question, Han's conjecture and $τ$-Hochschild (co)homology (2509.05135 - Cibils et al., 5 Sep 2025) in Subsection 6.4: Does infinite global dimension imply infinite + or co+ global dimension?