What is the contribution, if any, to the cosmological constant of the omnipresent static non-zero Higgs field? Are there 2 vacuum catastrophes?

The Standard Model requires that the universe is filled with a non-vanishing static Higgs field. This should give a very, if not unreasonably large contribution to the cosmological constant, competing with the QFT zero point energy aka 'the worst theoretical prediction in the history of physics.'. Experimentally the cosmological constant is fairly modest, about 0.2 proton mass per m$^3$. What is the mainstream explanation of this paradox? Do we have two vacuum catastrophes?