  Gamma-ray emission during long-duration gamma-ray flare (LDGRF) events is
thought to be caused mainly by $>$300 MeV protons interacting with the ambient
plasma at or near the photosphere. Prolonged periods of the gamma-ray emission
have prompted the suggestion that the source of the energetic protons is
acceleration at a coronal mass ejection (CME)-driven shock, followed by
particle back-precipitation onto the solar atmosphere over extended times. We
study the latter hypothesis using test particle simulations, which allow us to
investigate whether scattering associated with turbulence aids particles in
overcoming the effect of magnetic mirroring, which impedes back-precipitation
by reflecting particles as they travel sunwards. The instantaneous
precipitation fraction, $P$, the proportion of protons that successfully
precipitate for injection at a fixed height, $r_i$, is studied as a function of
scattering mean free path, $\lambda$ and $r_i$. Upper limits to the total
precipitation fraction, $\overline{P}$, were calculated for eight LDGRF events
for moderate scattering conditions ($\lambda$=0.1 au). We find that the
presence of scattering helps back-precipitation compared to the scatter-free
case, although at very low $\lambda$ values outward convection with the solar
wind ultimately dominates. For eight LDGRF events, due to strong mirroring,
$\overline{P}$ is very small, between 0.56 and 0.93% even in the presence of
scattering. Time-extended acceleration and large total precipitation fractions,
as seen in the observations, cannot be reconciled for a moving shock source
according to our simulations. Therefore, it is not possible to obtain both long
duration $\gamma$ ray emission and efficient precipitation within this
scenario. These results challenge the CME shock source scenario as the main
mechanism for $\gamma$ ray production in LDGRFs.
