Unitization by folding finitely many infinite paths in a graph

Let a graph have all but finitely many vertices lying on finitely many infinite paths. Fold each of these paths so that it produces countably infinitely many edges between two distinct vertices. Folding unitization conjecture. The resulting graph algebra is a unitization of the original graph algebra. The conjecture concerns extending the finite-graph construction described above to graphs with finitely many infinite paths; the supplied text gives no resolution or further conditions.