  The fractional Chern insulators (FCIs) observed in pentalayer rhombohedral
graphene/hexagonal boron nitride superlattices have a unique origin contrary to
theoretical expectations: their non-interacting band structure is gapless,
unlike standard FCIs and the Landau level. Hartree-Fock (HF) calculations at
filling $\nu=1$ yield a gapped ground state with Chern number 1 through band
mixing, identifying a possible parent state. However, many-body calculations
restricted to the occupied HF band predispose the system towards FCIs and are
essentially uncontrolled. In this work, we use unbiased multi-band exact
diagonalization (ED) to allow fluctuations into the gapless bands for two
normal-ordering schemes. In the "charge neutrality" scheme, the weak moir\'e
potential leads to theoretical proposals based on Wigner crystal-like states.
However, we find that FCIs seen in 1-band ED calculations are destroyed by band
mixing, becoming gapless as fluctuations are included. In the "average" scheme,
the Coulomb interaction with the periodic valence charge background sets up a
stronger moir\'e potential. On small systems, FCIs at $\nu=1/3$ are destroyed
in multi-band calculations, while those at $\nu=2/3$ are initially
strengthened. However we do not converge to a stable FCI at $\nu=2/3$ even on
the largest accessible systems. These findings question prior results obtained
within projection to a single HF band. They suggest that current models do not
support FCIs with correlation length small enough to be converged in
accessible, unbiased ED calculations, or do not support FCIs at all.
