The indecomposability conjecture for projective functor images of simple modules

Let $S_n$ be the symmetric group, let $x,y\in S_n$, let $L_y$ be the corresponding simple highest weight module, and let $\theta_x$ be the projective functor associated with $x$. Indecomposability conjecture. The module $\theta_xL_y$ is either zero or indecomposable. This conjecture concerns the structure of modules obtained by applying projective functors to simple highest weight modules and is presented as an open problem in the supplied material, with no resolution evidence given.