  In this note we confirm the conjecture of Calegari, Garoufalidis and Zagier
in their recent paper in Ann. Sci. \'{E}cole Normale Sup. that
$R_\zeta=c_\zeta^2$, where $R_\zeta$ is their map on $K_3$ defined using the
cyclic quantum dilogarithm and $c_\zeta$ is the Chern class map on $K_3$.
