On an equivalence of divisors on ${\bar{M}}_{0,n}$ from Gromov-Witten theory and conformal blocks

We consider a conjecture that identifies two types of base point free divisors on ${\bar{M}}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simple Lie algebras in type A. Here we reduce this conjecture on ${\bar{M}}_{0,n}$ to the same statement for $n=4$. A reinterpretation leads to a proof of the conjecture on ${\bar{M}}_{0,n}$ for a large class, and we give sufficient conditions for the non-vanishing of these divisors.