Does infinite global dimension force infinite + and/or infinite co+ global dimension?

Ascertain whether every bound quiver algebra Λ of infinite global dimension necessarily has infinite + global dimension, infinite co+ global dimension, or both; equivalently, determine whether for each algebra Λ with infinite global dimension there exists a pair of vertices (y,x) such that yΛx ≠ 0 and either Tor^Λ_*(k_x,{}_yk) is infinite (for +) or Tor^Λ_*(k_y,{}_xk) is infinite (for co+).

Background

The paper introduces two notions for a bound quiver algebra Λ = kQ/I: infinite + global dimension (existence of vertices (y,x) with yΛx ≠ 0 and Tor^Λ_*(k_x,{}_yk) infinite) and infinite co+ global dimension (existence of (y,x) with yΛx ≠ 0 and Tor^Λ_*(k_y,{}_xk) infinite). These notions clearly imply infinite global dimension.

The authors point out that the converse direction—whether infinite global dimension implies infinite + and/or infinite co+ global dimension—is not known. This question is central to linking τ-Hochschild (co)homology with classical homological invariants since, in their main results, infinite + (resp. co+) global dimension is equivalent to infinite τ-Hochschild homology (resp. cohomology).