Structural balance theory is an established framework for studying social relationships of friendship and enmity. These relationships are modeled by a signed network whose energy potential measures the level of imbalance, while stochastic dynamics drives the network towards a state of minimum energy that captures social balance. It is known that this energy landscape has local minima that can trap socially-aware dynamics, preventing it from reaching balance. Here we first study the robustness and attractor properties of these local minima. We show that a stochastic process can reach them from an abundance of initial states, and that some local minima cannot be escaped by mild perturbations of the network. Motivated by these anomalies, we introduce Best Edge Dynamics (BED), a new plausible stochastic process. We prove that BED always reaches balance, and that it does so fast in various interesting settings.