The logarithmic-form cokernel bound for an Euler restriction

Let ${\mathcal{A}}$ be a hyperplane arrangement, let $H\in{\mathcal{A}}$, and write ${\mathcal{A}}'$ for the deletion and ${\mathcal{A}}^H$ for the restriction to $H$. Let

$$i_H:\Omega^1({\mathcal{A}}')\longrightarrow\Omega^1({\mathcal{A}}^H)$$

be the Euler restriction map, and let $LP({\mathcal{A}},H)=\chi_0({\mathcal{A}};|{\mathcal{A}}^H|-1)$. The logarithmic-form cokernel bound. The cokernel of $i_H$ is finite-dimensional and satisfies

$$\dim\operatorname{coker}i_H\leq LP({\mathcal{A}},H)=\chi_0({\mathcal{A}};|{\mathcal{A}}^H|-1).$$

The finite-dimensionality part is proved in the paper, while the stated upper bound is the remaining conjectural assertion.