The singleton-root conjecture for the Bernstein–Sato polynomial of a regular sequence

Let $X$ be the ambient space, let $f_1,\dots,f_r\in \mathscr{O}_X(X)$ be a regular sequence, and let $g$ and $U$ be as in the preceding construction. Write $b_g(s)$ for the Bernstein–Sato polynomial of $g$ and $b_{g\vert_U}(s)$ for its restriction to $U$. Let $I$ denote the set of nonzero indices corresponding to the additional factors $(s+r+i)$ in the factorization of $b_g(s)$. Singleton-root conjecture. One has

$$b_g(s)= \begin{cases} b_{g\vert_U}(s)(s+r),\\ b_{g\vert_U}(s). \end{cases}$$

Equivalently, the set $I$ is either empty or the singleton $\{0\}$. The conjecture would show that no factor $(s+r+j)$ with $j>0$ can occur without the corresponding additional integer root phenomenon in the restricted Bernstein–Sato polynomial; the surrounding discussion motivates this from the nonvanishing of the quotient piece, but the text gives no resolution.