AMU conjecture on detecting pseudo-Anosov mapping classes by TQFT representations

Let $\Sigma$ be the surface with mapping class group $\Gamma$, and let $\rho_k$ denote the level-$k$ TQFT representation of $\Gamma$. A mapping class is Pseudo-Anosov if it is pseudo-Anosov, and it is reducible with pseudo-Anosov pieces if its canonical reduction has pseudo-Anosov components.

AMU conjecture. For a mapping class $\phi\in\Gamma$, $\phi$ is Pseudo-Anosov or reducible with Pseudo-Anosov pieces if and only if $\rho_k(\phi)$ is infinite order for large enough $k$.

This conjecture proposes that the asymptotic order of TQFT representations detects precisely the mapping classes having pseudo-Anosov dynamics on at least one component. The supplied text presents it as an expectation, and gives no resolution.