Regularity and strong regularity on Fano varieties

I will recall the notion of regularity for pluri-anticanonical divisors on Fano varieties, introduced by V.V. Shokurov. This invariant helps in studying the geometry of such varieties. A known result states that in the case of maximal regularity, it is achieved either on 1- or 2-complements. In joint work with J. Liu, we introduce the notion of strong regularity, for which the following holds: if strong regularity is maximal, then it is achieved on a 1-complement. In particular, a pair of maximal strong regularity is 1-complementary.