Semistable reduction conjecture

Let $R$ be a henselian discrete valuation ring with fraction field $K$ and residue field $k$ of characteristic $p>0$, and let $S=\operatorname{Spec}(R)$. For a proper morphism $f:X\to S$, write $X_\eta$ for its generic fibre over $\eta=\operatorname{Spec}(K)$. A proper model over the normalization $S'$ of $S$ in a finite extension $\eta'$ of $\eta$ is semistable when it has the local form $S'[t_1,\ldots,t_n]/(t_1\cdots t_r-\pi')$ for a uniformizer $\pi'$ of the valuation ring of $\eta'$. Semistable reduction conjecture. If $X_\eta$ is smooth over $\eta$, then there \exists a finite extension $\eta'$ of $\eta$ such that $X_{\eta'}$ admits a proper semistable model $X'\to S'$ on the normalization $S'$ of $S$ in $\eta'$. This is a fundamental semistable-reduction assertion, known in the source at least when the generic fibre is a smooth projective curve; the general case is presented as conjectural.