Azevedo–Shumyatsky conciseness conjecture for commutators of non-commutator words

Let $u_1,\ldots,u_r$ be non-commutator words in pairwise disjoint sets of variables. For a group word $v=v(x_1,\ldots,x_r)$, write $v(u_1,\ldots,u_r)$ for the word obtained by substituting the $u_i$ into $v$. A word is concise if, in every group, finiteness of its set of values implies finiteness of its verbal subgroup.

Azevedo–Shumyatsky conjecture. The word

$$\gamma_r(u_1,\ldots,u_r)=[u_1,\ldots,u_r]$$

is concise.

This conjecture generalizes known conciseness results for commutators involving non-commutator words; the paper's abstract states that it proves this generalized conjecture for lower central and derived words.