The virtual mapping-torus conjecture for negative immersions

A compact $2$-complex $X$ has negative immersions when it satisfies the negative-immersion property defined in the source. An immersion is a map $\psi:H\rightarrow F$ between graphs, and its finite directed height is the finiteness condition on directed intersections associated with the immersion. The virtual mapping-torus conjecture for negative immersions. If $X$ is a compact $2$-complex with negative immersions, then $\pi_1X$ has a finite index subgroup isomorphic to the fundamental group of a mapping torus of an immersion $\psi:H\rightarrow F$ of finite directed height. This conjecture proposes that negative immersions force a virtual mapping-torus structure; the source gives no resolution.