Orlik–Randell conjecture on cohomology of generic arrangement Milnor fibers

Let $Q=H_1\cdots H_k$ be a reduced polynomial defining a central generic arrangement in $\mathbb C^n$, let $R_n$ be the polynomial ring, let $E$ be the vector space defined by the displayed Jacobian expressions in the source, and let $\omega$ and $\pi$ denote the differential form and map used in the Orlik–Randell construction. Write $U_r$ for the degree-$r$ graded piece of $U$.

Orlik–Randell conjecture. For the fiber $X_1=\operatorname{Var}(Q-1)$, there is a finite-dimensional homogeneous vector space $U\subset R_n$ such that

$$R_n=E\oplus (\mathbb C[Q]\otimes U),$$ $$U\longrightarrow H^{n-1}(X_1,\mathbb C),\qquad g\longmapsto \bar{\pi(g\omega)}$$

is an isomorphism, and

$$\Omega^{n-1}_\alpha=\pi(U\omega)\oplus d\Omega^{n-2}_\alpha.$$

Moreover,

$$\nu_r=\begin{cases}\displaystyle {r+n-1\choose n-1},&0\le r\le k-n,\\\\displaystyle {k-2\choose n-1},&k-n+1\le r\le k-1,\\\\displaystyle {k-2\choose n-1}-{r-k+n-1\choose n-1},&k\le r\le 2k-n-2.\end{cases}$$

This conjecture proposes a homogeneous description of representatives for the Milnor-fiber cohomology, analogous to the isolated-singularity case; the source gives no resolution of the full assertion.