Semistable reduction conjecture for abelian varieties and dynamical systems

Let $K$ be a complete discretely valued field. An extension $L/K$ is weakly totally ramified if it is separable and has the relevant ramification property while preserving the residue field. Let $A_{/K}$ be an abelian variety, and let $f\colon \PP^n_K\to\PP^n_K$ be a dynamical system of degree $d\geq 2$. Semistable reduction conjecture. There is a separable weakly totally ramified extension $L/K$ and a model for $A_L$ that admits semistable reduction; likewise, there is such an extension $L/K$ and a model for $f_L$ that admits semistable reduction. The preceding discussion explains that Ramified Approximation provides this kind of result for elliptic curves and dynamical systems on $\PP^1_K$, while extensions to general abelian varieties and dynamical systems on higher-dimensional projective space are unclear; the quasi-finite residue-field case offers some partial motivation but does not establish the assertions.