Irreducible 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.

Irreducible curvatures of mapping tori of free group endomorphisms. Then $\rho_+(X)=0$.

The lower bound $\rho_+(X)\geq 0$ follows from $\chi(X)=0$. The source notes Wise's weaker non-positive immersions result and suggests that adapting its argument may prove this conjecture.