On two-generator subgroups of mapping torus groups

We prove that if $G_{\phi }=⟨F,t|tx{t}^{-1}=\phi \left(x\right),x\in F⟩$ is the mapping torus group of an injective endomorphism $\phi :F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_{\phi }$ is either free or a (finitary) sub-mapping torus. As an application we show that if $\phi \in \text{Out}\left(F_r\right)$ is a fully irreducible atoroidal automorphism, then every two-generator subgroup of $G_{\phi }$ is either free or has finite index in $G_{\phi }$.