Indecomposability conjecture for the modules $\widehat{V}$ and $\widehat{W}$

Let $\widehat{\mathfrak{b}}$ be the algebra whose modules are under consideration, and let $\widehat{V}$ and $\widehat{W}$ be the $\widehat{\mathfrak{b}}$-submodules appearing in the decomposition

$$\widetilde{V}(\mathbf{M},\mathbf{s},c_L,c_M)\cong\widehat{V}\oplus\widehat{W}$$

of the massive module under the condition

$$\mathbf{M}+\frac{n^2-1}{24}c_M\neq 0$$

for every positive integer $n$. Indecomposability conjecture. The $\widehat{\mathfrak{b}}$-modules $\widehat{V}$ and $\widehat{W}$ are indecomposable. This conjecture concerns the internal structure of the two summands in the decomposition of the massive module; their indecomposability remains an avenue for further study.