Violation of finite approximability for single-coefficient commutant equations

We construct an equation (resolved with respect to unknowns) such that (i) it has no solution in $F_2$ (a free group of rank $2$), but (ii) it has a solution in any finite homomorphic image of $F_2$. The left-hand-side of this equation belongs to the derived subgroup (i.e. has zero sum of exponents in each variable), while its right-hand-side is the commutator of two generators of $F_2$.