The non-annihilating graph conjecture for Monster-type axial algebras

Let $A$ be an axial algebra of Monster-type with generating axes $X$, and let $\Delta(X)$ be the graph whose vertices are the axes in $X$, with distinct vertices $a,b$ adjacent exactly when $ab\neq 0$. A sum decomposition is a decomposition $A=\sum_{i\in I}A_i$ into axial subalgebras, with distinct summands annihilating one another, and let $X_i\subseteq X$ be the axes contained in $A_i$. Non-annihilating graph conjecture. The finest sum decomposition of $A$ arises when each $X_i$ is a single connected component of $\Delta(X)$. This conjecture proposes that the connected components of the non-annihilating graph determine the finest sum decomposition; the supplied text gives no resolution status.