The recent work [Kurz et al., Numer. Math., 147 (2021)] proposed functional a posteriori error estimates for boundary element methods (BEMs) together with a related adaptive mesh-refinement strategy. Unlike most a posteriori BEM error estimators, the proposed functional error estimators cover Galerkin as well as collocation BEM and, more importantly, do not control the error in the integral density on the boundary, but the error of the potential approximation in the domain, which is of greater relevance in practice. The estimates rely on the numerical solution of auxiliary problems on auxiliary strip domains along the boundary, where the strips are affected by the adaptive mesh-refinement and hence vary. For Galerkin BEM, we prove that the proposed adaptive mesh-refinement algorithm yields convergence of the potential error to zero. Due to the structural difference to residual-based estimators, the proof requires new ideas.