Surface curvatures of mapping tori of free group endomorphisms

Let $X$ be a compact mapping torus of an injective endomorphism of a finitely generated free group.

Surface curvatures of mapping tori of free group endomorphisms. Then $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

The source calls this the most optimistic possible statement, while warning that its validity depends on choosing the appropriate 2-complex model; no resolution is given.