The algebraic generation conjecture for the homology of $d_W$

Let $W$ be a potential and let $d_W$ be the differential determined by

$$d_W(\xi(z))=\partial W(u(z))\mod z^n,$$

where $u(z)=\sum_{i=0}^{n-1}u_i z^i$ and $\xi(z)=\sum_{i=0}^{n-1}\xi_i z^i$. Define

$$H(x)=\operatorname{GCD}(\partial W(x),\partial^2 W(x))$$

and

$$\mu^W(z)=\sum_{i=0}^{n-1}\mu_i^Wz^i=\frac{\partial^2 W(u(z))}{H(u(z))}\dot{u}(z)\xi(z)-\frac{\partial W(u(z))}{H(u(z))}\dot{\xi}(z)\mod z^n.$$

Algebraic generation conjecture. The homology of $d_W$ is generated by $u_i$ and $\mu_k^W$ as an algebra. The supplied text establishes that the coefficients $\mu_i^W$ are cycles, but gives no resolution status for this generation claim.