Indecomposability conjecture for disconnected regular Schur labeled skew shape modules

Let $P\in\sfRSP_n$ be a regular Schur labeled skew shape poset, and let $\sh(\tau_P)$ denote its associated skew shape. A disconnected ribbon is a ribbon contained in a disconnected skew shape in the sense used in the paper.

Indecomposability conjecture. If $\sh(\tau_P)$ is disconnected and contains no disconnected ribbon, then the associated module $\sfM_P$ is indecomposable.

The converse to the paper's obstruction result is known when $\sh(\tau_P)$ is connected and was verified computationally for $|P|\leq 6$ in the disconnected case. The conjecture asserts the general disconnected case.