A method and system for controlling connectivity of a user equipment device (UE), the UE being dual-connectivity capable. The UE detects that the UE has ping-ponged at a threshold high rate between standalone connectivity and dual connectivity. And responsive to at least detecting that the UE has ping-ponged at the threshold high rate between standalone connectivity and dual connectivity, the UE transmits, to an access node serving the UE, a report including an indication that the UE is not dual-connectivity capable, even though the UE is dual-connectivity capable. The access node could then use the indication in the report as a basis to forgo configuring of dual connectivity for the UE, which could thereby help discontinue the ping-ponging.