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 $\Gamma_i$. 

Does there exist a smooth curve over $X/\mathbb{Q}$, such that every $\Gamma_i$ appear as the reduction of certain prime $p_i$?