  The nature of the three narrow hidden-charm pentaquark $P_c$ states is under
intense discussion since their discovery from the updated analysis of the
process $\Lambda_b^0\to J/\psi pK^-$ by LHCb. In this work we extend our
previous coupled-channel approach [Phys. Rev. Lett. 124, 072001 (2020)] by
including the $\Lambda_c\bar{D}^{(*)}$ and $\eta_cp$ explicitly as required by
unitarity and HQSS, respectively. Three calculation schemes are considered: (a)
scheme I with pure contact interactions between the elastic and inelastic
channels and without the $\Lambda_c\bar D^{(*)}$ interactions, (b) scheme II,
where the OPE is added to scheme I, and (c) scheme III, where the
$\Lambda_c\bar D^{(*)}$ interactions are included in addition. It is shown that
to obtain cutoff independent results, OPE in the multichannel system is to be
supplemented with $S$-wave-to-$D$-wave mixing contact terms. As a result we
demonstrate that the experimental data are consistent with the interpretation
of the $P_c(4312)$ and $P_c(4440)/P_c(4457)$ as $\Sigma_c\bar{D}$ and $\Sigma_c
\bar{D}^{*}$ molecules, respectively, and that the data show clear evidence for
a new narrow state, $P_c(4380)$, identified as a $\Sigma_c^*\bar D$ molecule,
which should exist as a consequence of HQSS. However, now two solutions are
found in all schemes which describe the data equally well, and thus no
unambiguous conclusion about the quantum numbers of the $P_c(4440)$ and
$P_c(4457)$ from data in the $J/\psi p$ channel alone is possible. It is argued
that one of these solutions, in which the quantum numbers of the $P_c(4440)$
and $P_c(4457)$ are $J^P=3/2^-$ and $1/2^-$, respectively, is theoretically
preferred. Moreover, we demonstrate that the line shapes related to the
$P_c(4440)$ and $P_c(4457)$ in the $\Sigma_c^{(*)}\bar{D}$ and $\eta_cp$ mass
distributions will allow one to pin down their quantum numbers once the data
are available.
