The injectivity conjecture for finitely separated graph C*-algebras

Let $(E,C)$ be a finitely separated graph. Let $M(E,C)$ be the abelian monoid with generators $\{a_v\mid v\in E^0\}$ and relations

$$a_v=\sum_{e\in X}a_{r(e)}$$

for all $v\in E^0$ and all $X\in C_v$. There is a natural map from $M(E,C)$ to the monoid ${\mathcal V}(C^*(E,C))$ of Murray–von Neumann equivalence classes of projections.

Injectivity conjecture. The natural map

$$M(E,C)\to {\mathcal V}(C^*(E,C))$$

is injective.

This is one of two weaker conjectures introduced after the full monoid isomorphism conjecture, and is intended to be verifiable in situations of interest. The source does not establish it in general.