Krull–Schmidt property for categorifications of the BMW skein and q-Schur algebras

Let $\mathcal{X}'$ be the additive category obtained from the categorification construction by adjoining every object $A$ of the ambient category $\mathcal{Y}$ for which there exist objects $B,C$ in $\mathcal{X}$ with $A\oplus B\cong C$. Let $\operatorname{Skein}_q(n,N)$ and $S_q(n,\Delta_N)$ denote the skein algebra and the corresponding $q$-Schur algebra appearing in the construction.

Krull–Schmidt property. In the case of the categorifications of $\operatorname{Skein}_q(n,N)$ and $S_q(n,\Delta_N)$, the category $\mathcal{X}'$ has the Krull–Schmidt property.

The Krull–Schmidt property is desirable because it ensures that objects decompose into indecomposable objects in an essentially unique way. The surrounding discussion reports that this property was proved by hand for the cases up to $\operatorname{BMW}_q(3,N)$, while the stated claim concerns the skein and $q$-Schur categorifications. The parser supplies no resolution status, so the claim is recorded as open.