The fibration conjecture for composed Schubert problems

Let $\underline{\lambda}$ and $\underline{\rho}$ be nontrivial Schubert problems, and let $\underline{\lambda}\circ\underline{\rho}$ be their composition. A composition is fibered over $\underline{\lambda}$ with fiber $\underline{\rho}$ if, for general instances, its solutions correspond bijectively to pairs consisting of a solution of $\underline{\lambda}$ and a solution of an instance of $\underline{\rho}$ depending on that solution, with the generality conditions specified in the definition of fibration. Fibration conjecture. A composed Schubert problem $\underline{\lambda}\circ\underline{\rho}$ is fibered over $\underline{\lambda}$ with fiber $\underline{\rho}$. By the preceding lemma, this stronger conjecture would imply that the Galois group of every nontrivial composition is imprimitive. The paper presents it as open and gives computational evidence for the weaker imprimitive-Galois-group conjecture.