Abstract: The Wilson coefficients of the standard model effective field theory are subject to a series of positivity bounds. It has been shown that while the positivity part of the ultraviolet (UV) partial wave unitarity leads to the Wilson coefficients living in a convex cone, further including the nonpositivity part caps the cone from above. For Higgs scattering, a capped positivity cone was obtained using a simplified, linear unitarity condition without utilizing the full internal symmetries of Higgs scattering. Here, we further implement stronger nonlinear unitarity conditions from the UV, which generically gives rise to better bounds. We show that, for the Higgs case in particular, while the nonlinear unitarity conditions per se do not enhance the bounds, the fuller use of the internal symmetries do shrink the capped positivity cone significantly.