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

Determine whether there exists a bound quiver algebra Λ of infinite global dimension that is not of infinite co+ 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_y,{}_xk) is finite (i.e., non-infinite).

Background

The authors likewise consider the symmetric implication for co+ global dimension. A counterexample to the implication “infinite global dimension ⇒ infinite co+ global dimension” would be a negative answer to Happel’s question within the non-local setting, but the authors are unaware of such an example.

This yields a parallel existence problem to the + case, now for the co+ condition.