The Watanabe–Yoshida conjecture states that the Hilbert–Kunz multiplicity, an invariant native to local rings in positive characteristic, attains its minimal value across singularities at the quadratic forms. Further, it claims that quadrics are characterised by attaining this minimum, but this part of the conjecture remained largely unaddressed in the literature.

In this talk, I will present an affirmative answer to this stronger form of the conjecture for complete intersection singularities in every positive characteristic, building on results by Enescu and Shimomoto, and using techniques by Han and Monsky. As a corollary, thanks to the progress on the first part of the conjecture up to dimension 7, also the stronger part follows in low-dimension.