Convergence Choquet-complete spaces and domain representations

de Brecht, Goubault-Larrecq, Jia and Lyu asked whether every sober convergence Choquet-complete space is domain-complete. We introduce the notion of singleton Choquet-completeness, a weakening of convergence Choquet-completeness in which the open sets chosen by player $\alpha$ are required to have a singleton intersection, but not necessarily to form a neighbourhood basis. We prove that every singleton Choquet-complete $T_1$ space is domain-representable. Consequently, every convergence Choquet-complete $T_1$ space is domain-representable and hence sober. Thus, in the $T_1$ case, the sobriety assumption in the above question is redundant, and the question reduces to whether every convergence Choquet-complete $T_1$ space is domain-complete.