Degree bound conjecture for Bernstein–Sato polynomials of homogeneous polynomials

Let $P(x_1,\ldots,x_n)\not\equiv0$ be a homogeneous polynomial in $n$ variables with coefficients in a field $K$ of characteristic $0$, and let $b(s)$ be its Bernstein–Sato polynomial. Degree bound conjecture. One has

$$\deg b\geq\deg P.$$

This would give a general lower bound on the degree of a Bernstein–Sato polynomial in the homogeneous case. The paper notes that the bound can fail for nonhomogeneous polynomials and that examples with $\deg b>\deg P$ exist even for homogeneous polynomials, but reports no counterexample to the conjecture.