Solving equations in SO(3) : an open problem by Jan Mycielski

Let me collect a number of known results:

i) $\alpha$ can be arbitrarily small, see my paper

Andreas Thom, Convergent sequences in discrete groups, Canad. Math. Bull. 56 (2013), no. 2, 424–433.

ii) There is some interest in estimating how small $\alpha$ can be in terms of the word length, this has been studied in the paper above, but also in

Abdelrhman Elkasapy and Andreas Thom, On the length of the shortest non-trivial element in the derived and the lower central series. J. Group Theory 18 (2015), no. 5, 793–804.

iii) In many cases $\alpha= \pi$. This has been studied in

Abdelrhman Elkasapy and Andreas Thom, About Gotô's method showing surjectivity of word maps, Indiana Univ. Math. J. 63 (2014), no. 5, 1553–1565.

and a more recent preprint

Anton Klyachko and Andreas Thom, New topological methods to solve equations over groups, arXiv:1509.01376

iv) The shortest word that I know for which $\alpha< \pi/2$ is $w=[[[XY X,YXY],[XYX,Y]],[[XYYX,YXXY],[X,Y]]]$

I'll post this as an answer so the question can be marked appropriately.

Corollary 3.3 in the paper Vladimir linked in the comments (arxiv.org/abs/1003.4093) says that $\alpha$ can be arbitrarily small.