  Given a covering of a quiver (with potential), we show that the associated
Bridgeland stability scattering diagrams are related by a restriction operation
under the assumption of admitting a nice grading. We apply this to quivers with
potential associated to marked surfaces. In combination with recent results of
the second and third authors, our findings imply that the bracelets basis for a
once-punctured closed surface coincides with the theta basis for the associated
stability scattering diagram, and these stability scattering diagrams agree
with the corresponding cluster scattering diagrams of
Gross-Hacking-Keel-Kontsevich except in the case of the once-punctured torus.
