The Morita, singularity-category and shift-equivalence classification conjecture

Let $E$ and $F$ be finite regular graphs. Let $K_0^{\operatorname{gr}}$ be the graded Grothendieck group, let $C^*(E)$ and $C^*(F)$ be the graph $C^*$-algebras with their gauge actions, let $L(E)$ and $L(F)$ be the Leavitt path algebras, and set $A(E)=\mathsf{k}E/J^2$ and $A(F)=\mathsf{k}F/J^2$, where $J$ is the ideal generated by paths of positive length. Write $A_E$ and $A_F$ for the adjacency matrices, and $\operatorname{\mathbf D}_{\operatorname{sng}}$ for the singularity category.

Morita–singularity classification conjecture. The following conditions are equivalent:

1. There is an isomorphism of partially ordered $\mathbb{Z}[x,x^{-1}]$-modules

$$K_0^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_0^{\operatorname{gr}}(L(F)).$$

2. There is a gauge-preserving Morita equivalence between $C^*(E)$ and $C^*(F)$.
3. There is a graded Morita equivalence between $L(E)$ and $L(F)$.
4. The singularity categories $\operatorname{\mathbf D}_{\operatorname{sng}}(A(E))$ and $\operatorname{\mathbf D}_{\operatorname{sng}}(A(F))$ are triangulated equivalent.
5. The adjacency matrices $A_E$ and $A_F$ are shift equivalent.

The equivalence between graded Morita equivalence, derived equivalence and triangulated equivalence of singularity categories is known in the source; the displayed five-way equivalence is presented as a conjecture, linking these classifications with ordered graded $K$-theory and symbolic-dynamics shift equivalence.