Morita theory of finite representations of Leavitt path algebras

The Graded Classification Conjecture states that for finite directed graphs $E$ and $F$, the associated Leavitt path algebras $L_{\text{K}}(E)$ and $L_{\text{K}}(F)$ are graded Morita equivalent, i.e., $\text{Gr}{L}_{\text{K}}(E){\approx }_{\text{gr}}\text{Gr}{L}_{\text{K}}(F)$, if and only if, their graded Grothendieck groups are isomorphic $K_0^{\text{gr}}(L_{\text{K}}(E))\cong K_0^{\text{gr}}(L_{\text{K}}(F))$ as order-preserving $\mathbb{Z}[x,x^{-1}]$-modules. Furthermore, if under this isomorphism, the class $[L_{\text{K}}(E)]$ is sent to $[L_{\text{K}}(F)]$ then the algebras are graded isomorphic, i.e., $L_{\text{K}}(E){\cong }_{\text{gr}}{L}_{\text{K}}(F)$. In this note we show that, for finite graphs $E$ and $F$ with so sinks and sources, an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism $K_0^{\text{gr}}(L_{\text{K}}(E))\cong K_0^{\text{gr}}(L_{\text{K}}(F))$ gives that the categories of locally finite dimensional graded modules of $L_{\text{K}}(E)$ and $L_{\text{K}}(F)$ are equivalent, i.e., $\text{fGr}[\mathbb{Z}]L_{\text{K}}(E){\approx }_{\text{gr}}\text{fGr}[\mathbb{Z}]L_{\text{K}}(F)\mathrm{.}$ We further obtain that the category of finite dimensional (graded) modules are equivalent, i.e., $\text{fModd}{L}_{\text{K}}(E)\approx \text{fModd}{L}_{\text{K}}(F)$ and $\text{fGr}{L}_{\text{K}}(E){\approx }_{\text{gr}}\text{fGr}{L}_{\text{K}}(F)$.