  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 $LR$.
