# Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $$w$$ ower the alphabet of $$n$$ symbols $$\mathfrak{U_w}$$ is a variety of all groups $$G$$, that satisfy an identity $$\forall a_1, … , a_n \in G$$ $$w(a_1, … , a_n) = e$$. Is it true, that for any group word $$w$$ there exists a positive real number $$\epsilon (w) > 0$$, such that any finite group $$G$$ is in $$\mathfrak{U_w}$$ iff $$\frac{\lvert\{(a_1, … , a_n) \in G^n : w(a_1, … , a_n) = e\}\rvert}{{|G|}^n} > 1 – \epsilon(w)?$$

How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:

A finite group $$G$$ is abelian iff $$\frac{|\{(a, b) \in G^2 : [a, b] = e\}|}{{|G|}^2} > \frac{5}{8}.$$

This theorem can be rephrased using aforementioned terminology as $$\epsilon([a, b]) = \frac{3}{8}$$.

There also is a generalisation of this theorem, stating that a finite group $$G$$ is nilpotent of class $$n$$ iff $$\frac{|\{(a_0, a_1, … , a_n) \in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e\}|}{{|G|}^{n + 1}} > 1 – \frac{3}{2^{n + 2}},$$ thus making $$\epsilon([ … [[a_0, a_1], a_2]… a_n]) = \frac{3}{2^{n + 2}}$$.

However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .

Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.