2 added 52 characters in body

John, you know this already, and this is far from an answer, but I thought I'd say it here for the benefit of others who may want to think about the problem.

Call a word $w(x_1,x_2,\dots,x_k)$ "groupy" in the variable $x_i$ if, for fixed values of the other variables, the set of values of $x_i$ such that $w$ is the identity element is a subgroup of the whole group. Call $w$ "groupy" if it is groupy in all its inputs.

We can show that if $w$ is groupy, and $H \le G$, then $p_w(H) \ge p_w(G)$, giving the monotonically decreasing property on $p_w(G_n)$.

The word $x$ and the word $e$ are groupy for trivial reasons. Beyond these, the only groupy word I can think of is the commutator word $[x_1,x_2]$.

On the other hand, if we restrict ourselves to the variety of abelian groups, all words power words (e.g., $x^2$ or $x^3$) are groupy, hence the monotonically decreasing property holds.

The iterated commutator $[[x_1,x_2],x_3]$ is groupy in $x_3$ but not (in general) in $x_1$ or $x_2$ -- however, it is likely that the groupiness argument can be extended somewhat to cover these kinds of words too.

1

John, you know this already, and this is far from an answer, but I thought I'd say it here for the benefit of others who may want to think about the problem.

Call a word $w(x_1,x_2,\dots,x_k)$ "groupy" in the variable $x_i$ if, for fixed values of the other variables, the set of values of $x_i$ such that $w$ is the identity element is a subgroup of the whole group. Call $w$ "groupy" if it is groupy in all its inputs.

We can show that if $w$ is groupy, and $H \le G$, then $p_w(H) \ge p_w(G)$, giving the monotonically decreasing property on $p_w(G_n)$.

The word $x$ and the word $e$ are groupy for trivial reasons. Beyond these, the only groupy word I can think of is the commutator word $[x_1,x_2]$.

On the other hand, if we restrict ourselves to the variety of abelian groups, all words are groupy, hence the monotonically decreasing property holds.

The iterated commutator $[[x_1,x_2],x_3]$ is groupy in $x_3$ but not (in general) in $x_1$ or $x_2$ -- however, it is likely that the groupiness argument can be extended somewhat to cover these kinds of words too.