Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$?
By a Neron model, I mean a smooth model (not necessarily proper) with the "Neron universal property": for any smooth $O_K$-scheme $\mathcal Y$,
$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(Y_K).$$
Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Neron model? Does base change help? That is, does there exist a Neron model for $X_L$ after some suitable base change $L/K$?
I'm also interested in knowing if we can just have the nice property $\mathcal{X}(O_K) = X(K)$ for some smooth model $\mathcal X$ of $X$.
Of course, all of this stuff is clear if $X$ has good reduction over $O_K$.
I posted this question on MathOverflow as suggested in the comments:
https://mathoverflow.net/questions/110359/do-all-curves-have-neron-models