Presentation independence of quiver C*-algebras

Prove that the (pro-)\(\mathrm C^*\)-algebra \(C(Q,I,\theta)\) associated with a quiver algebra \(A=\mathbb C Q/I\) and stability parameter \(\theta\) depends only on \(A\) and \(\theta\), and not on the chosen presentation of \(A\) by generators and relations.

Background

The paper associates universal or pro-$\mathrm C^*$-algebras to quivers with relations by imposing both the algebraic relations and the stability-dependent moment-map relation. It leaves unresolved whether this construction is intrinsic to the algebra and stability parameter rather than dependent on the selected quiver presentation.