In Paper I [T. J. H. Hele and S. C. Althorpe, J. Chem. Phys. 138, 084108 (2013)] we derived a quantum transition-state theory (TST) by taking the t → 0+ limit of a new form of quantum flux-side time-correlation function containing a ring-polymer dividing surface. This t → 0+ limit appears to be unique in giving positive-definite Boltzmann statistics, and is identical to ring-polymer molecular dynamics (RPMD) TST. Here, we show that quantum TST (i.e., RPMD-TST) is exact if there is no recrossing (by the real-time quantum dynamics) of the ring-polymer dividing surface, nor of any surface orthogonal to it in the space describing fluctuations in the polymer-bead positions along the reaction coordinate. In practice, this means that RPMD-TST gives a good approximation to the exact quantum rate for direct reactions, provided the temperature is not too far below the cross-over to deep tunnelling. We derive these results by comparing the t → ∞ limit of the ring-polymer flux-side time-correlation function with that of a hybrid flux-side time-correlation function (containing a ring-polymer flux operator and a Miller-Schwarz-Tromp side function), and by representing the resulting ring-polymer momentum integrals as hypercubes. Together with Paper I, the results of this article validate a large number of RPMD calculations of reaction rates.