The LEF conjecture for nilpotent products

Let $k\geq 1$. A $k$-nilpotent product of groups is the quotient of their free product by the subgroup generated by commutators of weight $k+1$; a group is LEF if every finite subset embeds, with its partial multiplication, into a finite group. Consider a $k$-nilpotent product of a finite family of LEF groups.

The LEF nilpotent-product conjecture. For each $k\geq 1$, the $k$-nilpotent product of a finite family of LEF groups is LEF.

This is presented as a plausible extension of Golovin's theorem, which proves the analogous assertion for finite groups. The conjecture would extend the preceding LEF result for verbal wreath products from finite base groups to LEF base groups.