Direct-sum decomposition conjecture for the associated separable algebra

Fix a stable gapped domain wall $\CM$ between $\CC_1$ and $\CC_2$. Let $\CB$, $A_1$, $A_2$, and $\phi$ be as in the proposed condensation construction.

Separable-algebra decomposition conjecture. There is a separable algebra $B'\in\CM$ such that

$$\CB_{A_1}\boxtimes_{\CB}\phi\boxtimes_{\CB}\CB_{A_2}\simeq{}_{B'}\CM_{B'},$$

and $B'$ is a direct sum of indecomposable algebras:

$$B'\cong\bigoplus_i B_i.$$

This is presented as an algebraic algorithm for decomposing the wall into indecomposable 1d condensable algebras. No proof or resolution is given.