  Many theoretical models require powerful active galactic nuclei (AGNs) to
suppress star formation in distant galaxies and reproduce the observed
properties of today's massive galaxies. A recent study based on Herschel-SPIRE
submillimetre observations claimed to provide direct support for this picture,
reporting a significant decrease in the mean star-formation rates (SFRs) of the
most luminous AGNs (Lx>10^44 erg/s) at z=1-3 in the Chandra Deep Field-North
(CDF-N). In this letter we extend these results using Herschel-SPIRE 250um data
in the COSMOS and CDF-S fields to achieve an order of magnitude improvement in
the number of sources at Lx>10^44 erg/s. On the basis of our analysis, we find
no strong evidence for suppressed star formation in Lx>10^44 erg/s AGNs at
z=1-3. The mean SFRs of the AGNs are constant over the broad X-ray luminosity
range of Lx~10^43-10^45 erg/s (with mean SFRs consistent with typical
star-forming galaxies at z~2; <SFRs>~100-200 Msol/yr). We suggest that the
previous CDF-N results were likely due to low number statistics. We discuss our
results in the context of current theoretical models.
