Generating equivalence relations on graph C*-algebras by structure-preserving moves

Let the six relations indexed by $\mathsf{000}$, $\mathsf{001}$, $\mathsf{011}$, $\mathsf{100}$, $\mathsf{101}$, and $\mathsf{111}$ encode the corresponding notions of invariance for graph $C^*$-algebras. Let the moves be (O), (I+), (I-), (R+), (S), (C+), and (P+), and write $\langle\cdots\rangle$ for the equivalence relation generated by the indicated moves.

Generating-moves conjecture. In all six cases, the equivalence relation is generated by the moves that leave the graphs invariant in the relevant sense. Explicitly,

$$\overline{\underline{\mathsf{000}}}=\langle\text{(O)},\text{(I-)},\text{(R+)},\text{(S)},\text{(C+)},\text{(P+)}\rangle,$$ $$\overline{\underline{\mathsf{001}}}=\langle\text{(O)},\text{(I-)},\text{(R+)},\text{(S)}\rangle,$$ $$\overline{\underline{\mathsf{011}}}=\langle\text{(O)},\text{(I-)}\rangle,$$ $$\overline{\underline{\mathsf{100}}}=\langle\text{(O)},\text{(I+)},\text{(R+)},\text{(C+)},\text{(P+)}\rangle,$$ $$\overline{\underline{\mathsf{101}}}=\langle\text{(O)},\text{(I+)},\text{(R+)}\rangle,$$ $$\overline{\underline{\mathsf{111}}}=\langle\text{(O)},\text{(I+)}\rangle.$$

The paper presents partial generation theorems and invariance results as evidence, but the full generating property is not established; the surrounding discussion also notes open questions in the dynamical cases and possible connections with the Williams conjecture.