Boundary epsilon regularity for incompressible Navier--Stokes equations via weak-strong uniqueness

We show that finite-energy weak solutions to the incompressible Navier–Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L_t^4{L}_x^4$-norm of the solution is smaller than a constant depending only on the domain. This answers a problem raised in [D. Albritton, T. Barker, and C. Prange, J. Math. Fluid Mech. 25 (2023), Paper No. 49]. Our proof relies on a new slicing construction near the boundary of the domain.