  Recent work on consistent Kaluza-Klein reductions on Einstein-Sasaki spaces
prompted an intriguing conjecture that there might exist a consistent $S^5$
reduction of type IIB supergravity to give five-dimensional ${\cal N}=8$ gauged
supergravity coupled to a massive supermultiplet that includes the
breathing-mode scalar. Motivated by this, we investigate the possibility of
augmenting the usual ${\cal N}=8$ supergravity reduction to include a
breathing-mode scalar, and we show that this is in fact inconsistent. The
standard reduction to the massless ${\cal N}=8$ supermultiplet depends for its
consistency on a delicate interplay between properties of the ten-dimensional
type IIB theory and properties of the Killing vectors on $S^5$. Our
calculations show that turning on the breathing-mode is sufficient to destroy
the balance, and hence render the reduction inconsistent.
