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The variability caused by the proliferation of distributed energy resources (DERs) and the significant growth in unbalanced three-phase loads pose unprecedented challenges to distribution network operations. This paper focuses on how a distribution system operator (DSO), taking over the distribution grid and market operations, would develop a risk-aware flexibility market to mitigate uncertainties in an unbalanced three-phase power distribution network. First, a distributionally robust chance constraint (DRCC) method is devised to solve the unbalanced three-phase optimal power flow using a semidefinite programming (SDP) model. The DSO can apply the proposed solution to jointly clear energy and flexibility markets. Then, the DRCC model accuracy is improved by an information-sharing mechanism characterized by spatially-correlated uncertainties in the distribution grid. Further, a novel system-wide response function is derived to make the DRCC model tractable. Using the duality theory, the paper further investigates the physical composition of the DSO’s cleared flexibility prices to guide the unbalanced distribution network operation. Finally, the effectiveness of the risk-aware flexibility market is verified in a modified three-phase IEEE 34-node system. Results demonstrate that the flexibility market can quantify the impact of spatially correlated uncertainties and facilitate the utilization of flexible resources to mitigate uncertainties across the network.