Generalized Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms I: the figure eight knot complement

We propose a generalization of the Bonahon-Wong-Yang volume conjecture of quantum invariants of surface diffeomorphisms, by relating the asymptotics of the invariants with certain hyperbolic cone structure on the mapping torus determined by the choice of the invariant puncture weights. We prove the conjecture for the once-punctured torus bundle with the diffeomorphism given by the word $LR$.