Properties of the relative slope map

Determine the further properties of the relative slope map associated with two universally tight contact structures, including whether it defines a pseudo-distance on the fundamental group, whether the absolute values can be removed to obtain a quasimorphism in suitable circumstances, and whether it is related to representations of the fundamental group into Homeo(R).

Background

For two universally tight contact structures, the paper defines a relative slope map on the fundamental group and observes its homogeneity relation under powers, namely Sl_{\xi_1}(\gamman,\xi_2)=|n|Sl_{\xi_1}(\gamma,\xi_2). The authors ask whether this quantity has stronger geometric or algebraic properties, particularly in the R-covered setting, where they suggest that a quasimorphism-like behavior may occur.

References

What are the further properties? Does it define a pseudo-distance on $\pi_1(M)$? Are there circumstances where we can get rid of the absolute values and obtain a quasi-morphism on $\pi_1(M)$? It seems to be the case in the $R$-covered situation. Is this map related to representations of $\pi_1(M)$ in $Homeo (R)$?

— On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry  (2609.03700 - Bowden et al., 3 Sep 2026) in Section 5, Outlook: Further Questions, Question “Slopes and Quasimorphisms”