Let $K$ be a field complete for a discrete valuation. Assume that the residue field has characteristic $p > 0$. Let $A$ be an abelian variety over $K$ having the property that (for some prime $\ell \neq p$) the action of the absolute Galois group of $K$ on the $\ell$-adic Tate module $T_\ell(A)$ is tamely ramified.

One could hope that under this condition, there is a projective model for $A$ over $\mathcal{O}_K$ which is *log smooth*; here the log structure is the one induced by the special fibre. I am not sure whether this is known; does anyone know a reference for this?

If one can find a projective, semistable, Galois-equivariant model for the abelian variety after a tamely ramified base change, then I know how to go from there - but my argument uses heavy machinery... There is a nice paper by Künnemann (Duke 1998) in which he proves that one can find a projective semistable model after a finite base change. But it is not clear to me whether a tame base change is sufficient to obtain such a model if one assumes moreover that the Galois action on the Tate module is tamely ramified.