After nearly 50 years of research the Chern conjecture for isoparametric hypersurfaces in spheres is still an unsolved and important problem and in particular its local version is of great interest, since here one loses the power of Stokes' Theorem as a method for proving it. Here we present a related result for CMC hypersurfaces in $\mathbb{S}^{n+1}$ with constant scalar curvature and three distinct principal curvatures.