Let $g\geq3$ be an integer, let $\{\Gamma_i|i \in I\}$ be the set of all stable graphs of genus $g$. (We say a graph is stable if it is the dual graph of a stable curve.) 

Let $X$ be a curve defined over $\mathbb{Q}$, we say it has reduction type $\Gamma_i$ , if there is a model $\mathcal{X}_{p_i}$ over $\mathbb{Z}_{p_i}$, whose generic fiber is $X_{\mathbb{Q}_{p_i}}$ and special fiber is a stable curve with dual graph $\Gamma_i$. 

Given $g\geq3$, does there always exist a smooth curve over $X/\mathbb{Q}$, (or some number field $K$?) such that every $\Gamma_i$ appear as the reduction of certain prime $p_i$? 

(If we ask the same question over global function field (finite extension of $\mathbb{F}_p(t)$), I think the answer is "yes", as we can interpolate the loci in $\overline{\mathcal{M}}_g$ by curves.  )