Andersen–Masbaum–Ueno conjecture for mapping class groups

Let $\Sigma_{g,n}$ be a compact oriented surface, and let

$$\mathrm{Mod}(\Sigma_{g,n})=\mathrm{Homeo}^+(\Sigma_{g,n})/\mathrm{Homeo}_0(\Sigma_{g,n})$$

be its mapping class group, with homeomorphisms and isotopies fixed on the boundary. For an odd integer $r\geq 3$, let $\rho_r$ denote the direct sum of the $\mathrm{SO}_3$-Witten–Reshetikhin–Turaev representations associated to boundary colorings at a primitive $r$-th root of unity. Andersen–Masbaum–Ueno conjecture. For every $f\in \mathrm{Mod}(\Sigma_{g,n})$, the mapping class $f$ has a pseudo-Anosov part if and only if $\rho_r(f)$ has infinite order for all sufficiently large odd integers $r\geq 3$. The conjecture proposes a representation-theoretic criterion for detecting pseudo-Anosov components of mapping classes. Its status is not resolved in the supplied source.