Characteristic-polynomial equality for type B Shi and Ish arrangements

Let $G=([ll],E_G,L_G)$ be a digraph on $[ll]$ with loop set $L_G\subseteq[ll]$ and edge set $E_G\subseteq\{(i,j)\mid 1\leq i<j\leq\ell\}$. Define arrangements in $\mathbb{R}^{\ell}$ by

$$\mathcal{S}(G)=\operatorname{Cox}(B_{\ell})\cup\{x_i=1\mid i\in L_G\}\cup\{x_i-x_j=1\mid(i,j)\in E_G\},$$

and

$$\mathcal{I}(G)=\operatorname{Cox}(B_{\ell})\cup\{x_i=1\mid i\in L_G\}\cup\{x_i=\ell+2-j\mid(i,j)\in E_G\}.$$

Characteristic-polynomial equality conjecture. The arrangements $\mathcal{S}(G)$ and $\mathcal{I}(G)$ have the same characteristic polynomial for any digraph $G$. This conjecture extends the observed equality for the deleted type $B$ Shi and Ish arrangements beyond the cases where their associated cones are free; the stated equality is not established in the supplied text.