Existence of an infinitely mutable potential yielding spherical generation for every quiver

Determine whether for every finite quiver Q there exists at least one infinitely mutable potential W such that the Kontsevich–Soibelman critical cohomological Hall algebra Coha_{Q,W} is spherically generated, i.e., generated by the degree-one pieces corresponding to the simple dimension vectors.

Background

After presenting counterexamples to conjectures asserting spherical generation and W-independence for all infinitely mutable potentials, the paper proposes a weakened, existence-type modification. Rather than requiring spherical generation for every infinitely mutable potential, the modified goal is to find, for each quiver, at least one infinitely mutable potential that yields a spherically generated CoHA.

The authors indicate that Proposition 3.4 (on quasihomogeneous potentials and equality of graded dimensions with S_Q) provides a practical tool to paper this question and potentially verify spherical generation by comparing partition functions and graded dimensions.