The isomorphism conjecture for the Fomin–Kirillov algebra and the Nichols–Woronowicz algebra

Let ${\mathcal{E}}_n$ be the algebra generated by $[i,j]$ subject to the Fomin–Kirillov quadratic relations, let $V_{S_n}$ be the Yetter–Drinfeld module associated with the type $A_{n-1}$ root system, and let ${\mathcal{B}}(V_{S_n})$ be its Nichols–Woronowicz algebra. There is a surjective algebra homomorphism

$$\eta:{\mathcal{E}}_n\longrightarrow {\mathcal{B}}(V_{S_n}),\qquad [i,j]\longmapsto [e_i-e_j].$$

The isomorphism conjecture. The algebra homomorphism $\eta$ is an isomorphism. The statement appears as a conjecture in the source, but the supplied material gives no resolution evidence, so its status remains open.