  In earlier work, the authors introduced a conjecture which, for an
orientation-preserving diffeomorphism $\varphi \colon S \to S$ of a surface,
connects a certain quantum invariant of $\varphi$ with the hyperbolic volume of
its mapping torus $M_\varphi$. This article provides a proof of this conjecture
in the simplest case where it applies, namely when the surface $S$ is the
one-puncture torus and the mapping torus $M_\varphi$ is the complement of the
figure-eight knot.
