The Solomon–Terao palindromicity conjecture for free arrangements

Let ${\mathcal{A}}$ be a hyperplane arrangement in a vector space of dimension $\ell$, and let $\Psi({\mathcal{A}};x,t)$ denote its Solomon–Terao polynomial. A polynomial is palindromic when its coefficients are symmetric. The Solomon–Terao palindromicity conjecture. The arrangement ${\mathcal{A}}$ is free if and only if $\Psi({\mathcal{A}};x,-1)$ is palindromic; equivalently, there exist $d_1,\ldots,d_\ell$ such that

$$\Psi({\mathcal{A}};x,-1)=\prod_{i=1}^\ell \frac{1-x^{d_i+1}}{1-x}.$$

This is presented as a second open problem related to Solomon–Terao polynomials; the supplied text gives no resolution.