  The fraction of $\chi_{c1}$ and $\chi_{c2}$ decays in the prompt $J/\psi$
yield, $F_{\chi c}=\sigma_{\chi_c \to J/\psi}/\sigma_{J/\psi}$, is measured by
the LHCb detector in pPb collisions at $\sqrt{s_{NN}}=8.16$ TeV. The study
covers the forward ($1.5<y^*<4.0$) and backward ($-5.0<y^*<-2.5$) rapidity
regions, where $y^*$ is the $J/\psi$ rapidity in the nucleon-nucleon
center-of-mass system. Forward and backward rapidity samples correspond to
integrated luminosities of 13.6 $\pm$ 0.3 nb$^{-1}$ and 20.8 $\pm$ 0.5
nb$^{-1}$, respectively. The result is presented as a function of the $J/\psi$
transverse momentum $p_{T,J/\psi}$ in the range 1$<p_{T, J/\psi}<20$ GeV/$c$.
The $F_{\chi c}$ fraction at forward rapidity is compatible with the LHCb
measurement performed in $pp$ collisions at $\sqrt{s}=7$ TeV, whereas the
result at backward rapidity is 2.4 $\sigma$ larger than in the forward region
for $1<p_{T, J/\psi}<3$ GeV/$c$. The increase of $F_{\chi c}$ at low $p_{T,
J/\psi}$ at backward rapidity is compatible with the suppression of the
$\psi$(2S) contribution to the prompt $J/\psi$ yield. The lack of in-medium
dissociation of $\chi_c$ states observed in this study sets an upper limit of
180 MeV on the free energy available in these pPb collisions to dissociate or
inhibit charmonium state formation.
