The local splitting conjecture for very critical extensions

Let $D$ be the filtered $\varphi$-module under consideration, let $\sigma$ range over the relevant embeddings, and let $I$ range over subsets that are very critical for $\sigma$. Let $m$ be the multiplicity in the algebraic automorphic representation, let $m_{\sigma,I}$ be the multiplicities in the representation in the preceding extension, and let $\pi(D)$ be the associated locally analytic representation. The local splitting conjecture. One has $m_{\sigma,I}=m$ for every $(\sigma,I)$ such that $I$ is very critical for $\sigma$, and the extension in the preceding decomposition is split, hence equal to $\pi(D)^{\oplus m}$. This conjecture would imply the first part of the main conjecture under the Taylor--Wiles assumptions. The supplied text gives no resolution.