  Given a monomorphism $\Psi:\mathcal{H}\rightarrow \mathcal{F}$ where
$\mathcal{H}$ is a proper free factor of the free group $\mathcal{F}$, we show
the associated mapping torus $X$ of $\Psi$ has negative immersions iff
$\mathcal{H}$ has finite height in $\pi_1X$ iff $\Psi$ is fully irreducible. We
survey related properties and discuss possible directions to pursue further.
