Exchange-relation conjecture for cluster-category triangles

Let $H$ be a finite-dimensional hereditary algebra, let $\overline{T}$ be an almost complete basic tilting object of its cluster category $\operatorname{\mathcal C}$, and let $M,M^{\ast}$ be the complements of $\overline{T}$. Suppose there are triangles

$$M^{\ast}\longrightarrow B\longrightarrow M\longrightarrow M^{\ast}[1],$$ $$M\longrightarrow B'\longrightarrow M^{\ast\ast}\longrightarrow M[1].$$

Write $B=\coprod_{i\in I}B_i^{d_i}$ and $B'=\coprod_{j\in J}(B'_j)^{e_j}$, with pairwise non-isomorphic summands. Let $x,x'$ correspond to $M,M^{\ast}$, and let $x_i,x'_j$ correspond to $B_i,B'_j$.

Exchange-relation conjecture. The exchange relation in $\operatorname{\mathcal A}(H)$ is

$$xx'=\prod_{i\in I}x_i^{d_i}+\prod_{j\in J}(x'_j)^{e_j}.$$

In particular, $B$ and $B'$ should have no common direct summands.

The conjecture seeks to recover cluster exchange relations from the two complement triangles; the supplied text gives no resolution beyond supporting the proposed correspondence.