# Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $$M_U$$ is the mapping torus for a mapping class group element $$U\in \mathrm{Mod}(\Sigma_1)$$, then $$Z(M_U) = Z(M_{-U})$$ where $$-I\in \mathrm{Mod}(\Sigma_1)$$ is the hyperelliptic involution.

What about the genus 2 case? If $$-I\in \mathrm{Mod}(\Sigma_2)$$ is the hyperelliptic involution on the genus 2 surface, then is there any $$U\in \mathrm{Mod}(\Sigma_2)$$, for which the following does not hold? $$Z(M_U) \stackrel{?}{=} Z(M_{-U})$$