  We prove a certain uniform version of the Shafarevich Conjecture. As a
corollary, we prove the Rasmussen-Tamagawa Conjecture for a particular class of
abelian varieties $A$ defined over a number $K$ of dimension $g$ having
everywhere potential good reduction, in particular, for any finite place $v$ of
$K$ the localization $A_v:=A\times_{\mathrm{Spec}(K)}\mathrm{Spec}(K_v)$ has
either good reduction or {\it totally bad reduction} (connected component
$\tilde{\mathcal{A}}_v^0$ of the special fibre $\tilde{\mathcal{A}}_v$ of the
N\'eron model $\mathcal{A}_v$ at $v$ is an affine group scheme over the residue
field $k_v$ at $v$) and has good reduction over a quadratic extension of $K_v$.
