Schur–Baer property for all group words

Prove that every group word φ has the Schur–Baer property: for any group G, if the index |G : φ*(G)| is finite and equal to m, then the order of the verbal subgroup φ(G) is an m-number (i.e., an integer whose prime divisors lie among those of m). Here φ*(G) denotes the marginal subgroup of G corresponding to the word φ, consisting of all elements a ∈ G such that replacing any argument x_i of φ by x_i a leaves the value of φ unchanged for all choices of the remaining arguments in G.

Background

In Section 8 the authors define, for a group word φ, the associated verbal subgroup φ(G) and the marginal subgroup φ(G). A word φ is said to have the Schur–Baer property if, whenever |G : φ(G)| = m is finite, the order of φ(G) is an m-number. Schur’s classical theorem establishes this property for the commutator word γ₂.

Baer’s results in the same section show that the Schur–Baer property is preserved under certain operations (e.g., for commutator words formed from words that already have the property), and that it holds in polycyclic contexts. Despite these partial results, the notes explicitly conjecture the universal validity of the Schur–Baer property for all words, without restriction on the ambient group G or the form of φ.