The indecomposability conjecture for projective functors acting on simple modules

Let $\mathfrak{g}=\mathfrak{sl}_n(\mathbb{C})$, let $\mathcal{O}_0$ be the principal block of category $\mathcal{O}$, let $L$ be a simple object of $\mathcal{O}_0$, and let $\theta$ be an indecomposable projective functor on $\mathcal{O}_0$.

Indecomposability conjecture. The module $\theta L$ is either indecomposable or zero.

This conjecture is presented as an earlier approach toward the theorem on projective functors on parabolic category $\mathcal{O}$, but the supplied text gives no resolution status. A more precise reformulation appears later in the paper.