  We investigate and compare the particle number fluctuations in the putative
many-body localized (MBL) phase of a spinless fermion model with potential
disorder and nearest-neighbor interactions with those in the non-interacting
case (Anderson localization) and in effective models where only interaction
terms diagonal in the Anderson basis are kept. We demonstrate that these types
of simple effective models cannot account for the particle number fluctuations
observed in the MBL phase of the microscopic model. This implies that assisted
and pair hopping terms---generated when transforming the microscopic
Hamiltonian into the Anderson basis---cannot be neglected. As a consequence, it
appears questionable if the microscopic model possesses an exponential number
of exactly conserved local charges. If such exactly conserved local charges do
not exist, then particles are expected to ultimately delocalize for any finite
disorder strength.
