  QGP tomography aims to constrain the parameters characterizing the properties
and evolution of Quark-Gluon Plasma (QGP) formed in heavy-ion collisions, by
exploiting low and high-$p_\perp$ theory and data. Higher-order harmonics $v_n$
($n>2$) are an important -- but seldom explored -- part of this approach.
However, to take full advantage of them, several issues have to be addressed:
i) consistency of different methods for calculating $v_n$, ii) importance of
event-by-event fluctuations to high-$p_\perp$ $R_{AA}$ and $v_2$ predictions,
iii) sensitivity of higher harmonics to the initial state of fluid-dynamical
evolution. We obtain that i) several methods for calculating harmonics are
compatible with each other, ii) event-by-event calculations are important in
mid-central collisions, and iii) various initializations of the evolution of
the medium lead to quantitatively and qualitatively different predictions,
likely to be distinguished by future measurements. We also find that the
present high-$p_\perp$ $v_4$ data cannot be reproduced using initial states for
fluid-dynamical evolution given by state-of-the-art models. We call this
discrepancy high-$p_\perp$ $v_4$ puzzle at the LHC.
