The $A_n$-base conjecture for Mori conic bundles

Let $(X,C)$ be a germ of a three-dimensional complex space along a compact reduced curve $C$, let $(S,0)$ be a germ of a two-dimensional normal complex space, and suppose that $X$ has at worst terminal singularities. A proper morphism $f\colon (X,C)\to (S,0)$ is a Mori conic bundle if $(f^{-1}(0))_{\operatorname{red}}=C$, $f_*\mathcal{O}_X=\mathcal{O}_S$, and $-K_X$ is ample.

The $A_n$-base conjecture. If $f\colon (X,C)\to (S,0)$ is a Mori conic bundle, then $(S,0)$ is a Du Val singularity of type $A_n$.

The conjecture is relevant to the Sarkisov program, the rationality problem for conic bundles, and the study of $\mathbb{Q}$-Fano threefolds with extremal contractions onto surfaces. The Gorenstein case is known, as are Mori conic bundles whose points have indices at most $2$; the general case is presented as open here.