Multiplicity-free products of Schubert divisors

Let $G/B$ be a flag variety over $\mathbb C$, where $G$ is a simple algebraic group with a simply laced Dynkin diagram, and $B$ is a Borel subgroup. We say that the product of classes of Schubert divisors in the Chow ring is multiplicity free if it is possible to multiply it by a Schubert class (not necessarily of a divisor) and get the class of a point. In the present paper we find the maximal possible degree (in the Chow ring) of a multiplicity free product of classes of Schubert divisors.