A relation between the Baseilhac-Benedetti and the Bonahon-Liu-Wong-Yang invariants

Baseilhac-Benedetti, following ideas of Kashaev, introduced invariants of pseudo-Anosov homeomorphisms of punctured hyperbolic surfaces that depend on a complex root of unity of odd order. Around the same time, Bonahon-Liu introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. A little later, Dimofte and the first author introduced invariants of cusped hyperbolic 3-manifolds at roots of unity using their geometric representation. In another effort, Bonahon-Wong-Yang introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. All these invariants are conjecturally closely related, and our aim is to prove a precise relation between the Baseilhac-Benedetti invariants, the Bonahon-Liu-Wong-Yang and the lesser-known abelian $\mathfrak{gl}_1$-invariants.