  Hyper-order cumulants $C_5/C_1$ and $C_6/C_2$ of net-baryon distributions are
anticipated to offer crucial insights into the phase transition from
quark-gluon plasma to hadronic matter in heavy-ion collisions. However, the
accuracy of $C_5$ and $C_6$ is highly contingent on the fine shape of the
distribution's tail, the detectable range of which could be essentially
truncated by low statistics. In this paper, we use the fast Skellam-based
simulations, as well as the Ultrarelativistic Quantum Molecular Dynamics model,
to assess the impact of limited statistics on the measurements of $C_5/C_1$ and
$C_6/C_2$ of net-proton distributions at lower RHIC energies. Both ratios
decrease from the unity baseline as we reduce statistics, and could even turn
negative without a pertinent physics mechanism. By incorporating statistics
akin to experimental data, we can replicate the net-proton $C_5/C_1$ and
$C_6/C_2$ values comparable to the corresponding measurements for Au+Au
collisions at $\sqrt{s_{NN}} =$ 7.7, 11.5 and 14.5 GeV. Our findings underscore
a caveat to the interpretation of the observed beam energy dependence of
hyper-order cumulants.
