The Main Conjecture for the Bernstein–Sato function of the Vandermonde determinant

Let $\xi_n=\prod_{1\leq i<j\leq n}(x_i-x_j)$ be the Vandermonde determinant, and for an integer partition $\lambda\vdash n$, let $\xi_\lambda$ be the product of the Vandermonde determinants associated to the blocks of the corresponding set partition. Write $b_f(s)$ for the Bernstein–Sato $b$-function of $f$. Main Conjecture. The $b$-function of $\xi_n$ is given by the recursive formula

$$b_{\xi_n}(s)=\operatorname*{\lcm}_{\substack{\lambda\vdash n\lambda\neq(n)}}\left(b_{\xi_\lambda}(s)\right)\cdot\prod_{i=n-1}^{(n-1)^2}\left(s+\frac{i}{\binom{n}{2}}\right).$$

This conjecture gives a recursive description of the Bernstein–Sato function of the type $A_{n-1}$ Coxeter arrangement in terms of smaller set partitions. The paper proves a symmetry property of the relevant $\mathcal{D}$-modules and derives bounds toward the formula, but the stated equality remains conjectural here.