A half-century ago when simply elliptic singularity was introduced, it was a natural question whether its discriminant complement is a $K(\pi,1)$ space. At that time, Fulvio Lazzeri suggested the possibility of the existence of a non-trivial $\pi_2$ class by a heuristic argument on the real discriminant complement.
In the present talk, I approach this problem from elliptic Artin monoids, where the monoid is defined by generalizing the classical Artin braid relations to the new relations, called elliptic braid relations, defined on elliptic diagrams. Contrary to the classical Artin monoids, the elliptic Artin monoids are not cancellative and their natural homomorphisms to elliptic Artin groups (=the fundamental groups of the elliptic discriminant complements) are not injective (except for rank 1 case). This fact leads me to the construction of $\pi_2$-classes in the complement of the discriminant. We conjecture that they are non-vanishing.
Then, we reformulate the classical $K(\pi,1)$-conjecture for complements of discriminants, whether the discriminant complements are homotopic to classifying spaces associated to the elliptic Artin monoids.