The algebraic generation conjecture for the homology of $d_N$

Let $u_i$ and $\xi_j$ be the generators in the polynomial algebra underlying the differential $d_N$, and for $k=1,\ldots,n-1$ define

$$\mu_k=\sum_{i+j=k}(Ni-j)u_i\xi_j.$$

The differential satisfies $d_N(\xi(z))=u(z)^N\mod z^n$, where $u(z)=\sum_{i=0}^{n-1}u_i z^i$ and $\xi(z)=\sum_{i=0}^{n-1}\xi_i z^i$. Algebraic generation conjecture. The homology of $d_N$ is generated by $u_i$ and $\mu_k$ as an algebra. For $N=2$ and $n\leq 8$ this conjecture has been extensively verified, but it remains open in general.