  In a recent work [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 2024],
Gastel and Neff introduced an interesting system from a geometrically nonlinear
flat cosserat micropolar model and established interior regularity in the
critical dimension. Motived by this work, in this article, we establish both
interior regularity and sharp $L^p$ regularity for their system in
supercritical dimensions.
