Equivalence of $v$-decomposition matrices for blocks of Ariki-Koike algebras

We consider the representation theory of the Ariki-Koike algebra, a $q$-deformation of the group algebra of the complex reflection group $C_r \wr \mathfrak{S}_n$. We examine blocks of the Ariki-Koike algebra. In particular, we prove a sufficient condition such that restriction of modules leads to a natural correspondence between the multipartitions of $n$ whose Specht modules belong to a block $B$ and those of $n-\delta_i(B)$ whose Specht modules belong to the block $B'$, obtained from $B$ applying a Scopes' equivalence. This bijection gives us an equivalence for the $v$-decomposition numbers of the Ariki-Koike algebras.