  We present our Swift monitoring campaign of the slowly rotating neutron star
Be/X-ray transient GX 304-1 (spin period of ~275 s) when the source was not in
outburst. We found that between its type-I outbursts the source recurrently
exhibits a slowly decaying low-luminosity state (with luminosities of
10^(34-35) erg/s). This behaviour is very similar to what has been observed for
another slowly rotating system, GRO J1008-57. For that source, this
low-luminosity state has been explained in terms of accretion from a
non-ionised ('cold') accretion disk. Due to the many similarities between both
systems, we suggest that GX 304-1 enters a similar accretion regime between its
outbursts. The outburst activity of GX 304-1 ceased in 2016. Our continued
monitoring campaign shows that the source is in a quasi-stable low-luminosity
state (with luminosities a few factors lower than previously seen) for at least
one year now. Using our NuSTAR observation in this state, we found pulsations
at the spin period, demonstrating that the X-ray emission is due to accretion
of matter onto the neutron star surface. If the accretion geometry during this
quasi-stable state is the same as during the cold-disk state, then matter
indeed reaches the surface (as predicted) during this latter state. We discuss
our results in the context of the cold-disk accretion model.
