The freeness conjecture for the interpolating arrangements $\mathscr{D}^r_\ell(a)$

Let $a\geq 0$, $\ell\geq 1$, and $0\leq r\leq\ell$. Let $\mathscr{D}^r_\ell(a)$ be the arrangement consisting of the hyperplanes

$$2x_i=[-2a,0]\quad(1\leq i\leq r),\\ x_i+x_j=[-3a,a]\quad(1\leq i<j\leq r),\\ x_i-x_j=[-2a,2a]\quad(1\leq i<j\leq r),\\ x_i\pm x_j=[-2a,a]\quad(1\leq i\leq r<j\leq\ell),\\ x_i\pm x_j=[-a,a]\quad(r+1\leq i<j\leq\ell).$$

Freeness conjecture for $\mathscr{D}^r_\ell(a)$. The cone $\mathbf{c}\mathscr{D}^r_\ell(a)$ is free with exponents

$$\exp\left(\mathbf{c}\mathscr{D}^r_\ell(a)\right)=(1,\ell+r-1,1,3,5,\ldots,2\ell-3)+(0,(2a\ell+2ar-2a)^\ell).$$

The cases $r=0$ and $r=\ell$ are already established, and the conjecture is proposed to handle the remaining interpolating arrangements; its general status is open.