If there’s a high clinical suspicion (could be relevant in this case, though it may also represent the normal flu), the utility of the testing improves.
Looking at the three out of four FDA EUA-approved serology tests (see https://www.fda.gov/medical-devices/emergency-situations-med...) most do not report sensitivities up to 98%, with the notable exception of Ortho Vitro, though the diversity of the samples they test isn't described.
1. Cellex - https://www.fda.gov/media/136625/download
2. Ortho Vitros - https://www.fda.gov/media/136967/download
3. Chembio - https://www.fda.gov/media/136963/download