The Indecomposability Conjecture for projective functors in category O

Let $\theta_x^{(n)}$ be the projective functor indexed by $x\in\rm S_n$, and let $L_z^{(n)}$ be the simple highest weight module indexed by $z\in\rm S_n$ in the principal block of category $\mathcal O$. Write $\mathbf{KM}_n(x,z)=\mathtt{true}$ when $\theta_x^{(n)}L_z^{(n)}$ is zero or indecomposable, and let $\mathbf{KM}_n(\star,z)$ be the conjunction of these truth values over all $x\in\rm S_n$. Indecomposability Conjecture. For all $z\in\rm S_n$, we have $\mathbf{KM}_n(\star,z)=\mathtt{true}$. This conjecture concerns the preservation of indecomposability under projective functors and is stated as a conjecture first presented by Kildetoft and Mazorchuk; the paper verifies it for the permutation shapes under consideration.