The differential-value conjecture for Bernstein–Sato roots of cusps

Let $C$ be a cusp with semigroup

$$\Gamma_C=\langle n,m\rangle,$$

where $\gcd(n,m)=1$ and $2\leq n<m$. Let $\Lambda_C$ be the semimodule of differential values of $C$. For an implicit equation $f$ of $C$, let the Bernstein–Sato polynomial be the monic generator of the ideal of polynomials $B\in\mathbb C[\rho]$ for which some differential operator $P\in\mathcal D[\rho]$ satisfies

$$P(\rho)f^{\rho+1}=B(\rho)f^\rho.$$

The differential-value conjecture. For any element $\lambda\in\Lambda_C\setminus\Gamma_C$, the rational number $-\lambda/nm$ is a root of the Bernstein–Sato polynomial of $C$.

The claim extends the result proved in the paper for cusps with $n\leq 4$, and would identify all roots arising from differential values outside the value semigroup. Its status is not resolved in the supplied text.