Are primitive words universal for infinite symmetric groups?

Abstract:

Let $W = W({x_1}, \ldots ,{x_j})$ be any word in the $j$ free generators ${x_1}, \ldots ,{x_j}$, and suppose that $W$ cannot be expressed in the form $W = {V^k}$ for $V$ a word and for $k$ an integer with $\left | k \right | \ne 1$. We ask whether the equation $f = W$ has a solution $({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots , a_{j}) \in G^{j}$ whenever $G$ is an infinite symmetric group and $f$ is an element in $G$. We establish an affirmative answer in the case that $W(x,y) = {x^m}{y^n}$ for $m$ and $n$ nonzero integers.