Replacement theorem by characteristic subgroup satisfying a multilinear commutator identity

Statement
Suppose $$G$$ is a group and $$H$$ is a subgroup satisfying a multilinear commutator identity: in other words, there is a word $$w(x_1,x_2,\ldots,x_n)$$ composed entirely of iterated commutation, such that $$w(x_1,x_2,\ldots,x_n) = e$$ for all $$x_i \in H$$. Then, there exists a characteristic subgroup of finite index $$N$$ of $$G$$ satisfying the same multilinear commutator identity, and where the index of $$N$$ is bounded from above by a function of $$n$$ and $$w$$ (independent of $$G$$).

Particular cases

 * If $$H$$ is nilpotent of class $$c$$, we can find a characteristic subgroup of finite index, also nilpotent of the same class $$c$$, and with index bounded from above by a function of $$n$$ and $$c$$.
 * If $$H$$ is solvable with derived length $$l$$, we can find a characteristic subgroup of finite index, also solvable with derived length $$l$$, and with index bounded from above by a function of $$n$$ and $$l$$.