Injectivity conjecture for the natural maps associated with the mathbb Z_3-symmetric down-up algebra

Let the maps in Proposition and Theorem be the natural maps associated with the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A=\mathbb A(\alpha,\beta,\gamma)$. Injectivity conjecture. The maps in Proposition and Theorem are injective. This conjecture concerns the faithfulness of the natural algebraic constructions introduced earlier in the paper; the supplied context does not state whether it has been resolved.