  We present new Chandra X-ray observations of seven low-mass black holes (~1e6
Msun) accreting at low Eddington ratios between -2.0<log L/Ledd<-1.5. We
compare the X-ray properties of these seven low-mass active galactic nuclei
(AGN) to a total of 73 other low-mass AGN in the literature with published
Chandra observations (with Eddington ratios extending from -2.0<log
L/Ledd<-0.1). We do not find any statistical differences between low- and
high-Eddington ratio low-mass AGN in the distributions of their X-ray to
ultraviolet luminosity ratios (aox), or in their X-ray spectral shapes.
Furthermore, the aox distribution of low-L/Ledd AGN displays an X-ray weak tail
that is also observed within high-L/Ledd objects. Our results indicate that
between -2<log L/Ledd<-0.1, there is no systematic change in the structure of
the accretion flow for active galaxies hosting 1e6 Msun black holes. We examine
the accuracy of current bolometric luminosity estimates for our low-L/Ledd
objects with new Chandra observations, and it is plausible that their Eddington
ratios could be underestimated by up to an order of magnitude. If so, then in
analogy with weak emission line quasars, we suggest that accretion from a
geometrically thick, radiatively inefficient `slim disk' could explain their
diverse properties in aox. Alternatively, if current Eddington ratios are in
fact correct (or overestimated), then the X-ray weak tail would imply that
there is diversity in disk/corona couplings among individual low-mass objects.
Finally, we conclude by noting that the aox distribution for low-mass black
holes may have favorable consequences for the epoch of cosmic reionization
being driven by AGN.
