Equivalence-relation conjecture for projective-functor linkage

Let $L$ be a simple $\mathfrak{g}$-module, and let $\mathcal{X}_L$ be the class of simple modules linked to $L$ by projective functors. Define $M\triangleright N$ when $M$ occurs in the top or socle of a module obtained from $N$ by applying a projective functor. Equivalence-relation conjecture. The relation $\triangleright$ is an equivalence relation. The paper proves a theorem showing that equivalence in one nonempty central-character component propagates to all components, and uses this in a special setup; the general assertion is not resolved by the supplied context.