  We reconsider the long-explored problem that the magnitude of the measured
Casimir force between an Au sphere and an indium tin oxide (ITO) film decreases
significantly with no respective changes in the ITO dielectric permittivity
required by the Lifshitz theory. Two plausible resolutions of this conundrum
are discussed: the phase transition of an ITO film from metallic to dielectric
state and the modification of a film surface under the action of UV light. To
exclude the latter option, we propose an improvement in the experimental scheme
by adding a graphene sheet on top of an ITO film. The formalism is developed
allowing precise calculation of the Casimir force between an Au sphere and a
graphene sheet on top of ITO film deposited on a quartz substrate. In doing so
Au, ITO, and quartz are described by the frequency-dependent dielectric
permittivities and real graphene sheet with nonzero mass-gap parameter and
chemical potential by the polarization tensor at nonzero temperature. Numerical
computations performed both before and after the phase transition resulting
from the UV treatment show that the presence of graphene leads to only a minor
decrease in the drop of the Casimir force which remains quite measurable. At
the same time, in the presence of graphene the guess that an observed drop
originates from the modification of an ITO surface by the UV light breaks down.
Similar results are obtained for the configuration of two parallel plates
consisting of a graphene sheet, an ITO film and a quartz substrate. The
proposed experiments involving additional graphene sheets may help in
resolution of the problems arising in application of the Lifshitz theory to
real materials.
