I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/2321255).

Let $p_1, q_1,...., p_m, q_m$ be fixed integers (both positive or negative). I am interested in the image of the mapping $SO(3)\times SO(3) \to SO(3)$ given by $(X, Y)\mapsto X^{p_1} Y^{q_1} X^{p_2} Y^{q_2}\cdots X^{p_m}Y^{q_m}$. By conjugacy and continuity, it is easy to see that the image is the set of all rotations by angles in $[0,\alpha]$ for some $\alpha$ that depends on $p_1, q_1,...., p_m, q_m$. Since $SO(3)$ contains a copy of the free group on two generators, $\alpha$ is always strictly positive.

Mycielski asked whether $\alpha$ is always equal to $\pi$. I am intersted to know whether $\alpha$ is always at least $\pi/2$.

Is there anything new to be said about this problem, or is it still wide open? I checked the papers citing Mycielski's paper, but none of them seem to have a solution.