  We prove that the quantum DT invariants associated to quivers with genteel
potential can be expressed in terms of certain refined counts of tropical
disks. This is based on a quantum version of Bridgeland's description of
cluster scattering diagrams in terms of stabilitiy conditions, plus a new
version of the description of scattering diagrams in terms of tropical disk
counts. The weights with which the tropical disks are counted are expressed in
terms of motivic integrals of certain quiver flag varieties. We also show via
explicit counterexample that Hall algebra broken lines do not result in
consistent Hall algebra theta functions, i.e., they violate the extension of a
lemma of Carl-Pumperla-Siebert from the classical setting.
