Self-adaptation enables software systems to respond to changing environmental contexts that may not be fully understood at design time. Designing a dynamically adaptive system (DAS) to cope with this uncertainty is challenging, as it is impractical during requirements analysis and design time to anticipate every environmental condition that the DAS may encounter. Previously, the RELAX language was proposed to make requirements more tolerant to environmental uncertainty, and Claims were applied as markers of uncertainty that document how design assumptions affect goals. This paper integrates these two techniques in order to assess the validity of Claims at run time while tolerating minor and unanticipated environmental conditions that can trigger adaptations. We apply the proposed approach to the dynamic reconfiguration of a remote data mirroring network that must diffuse data while minimizing costs and exposure to data loss. Results show RELAXing Claims enables a DAS to reduce adaptation costs.