A Nielsen--Schreier variety of algebras without the PBW property

We prove the Nielsen–Schreier property for the variety of algebras defined by the identity $x(x^2{)}^2=(x^2{)}^{2}x$: every subalgebra of every free algebra in this variety is itself free. We also show that this variety of algebras does not have the Poincaré–Birkhoff–Witt property for universal multiplicative enveloping algebras. Our strategy of proof actually leads to infinitely many new varieties of non-associative algebras with the same behaviour. This offers new evidence supporting a conjecture of the first author and Umirbaev suggesting that the Nielsen–Schreier property over a field of zero characteristic is equivalent to freeness of universal multiplicative enveloping algebras of free algebras.