  We compute the in-medium jet broadening $\langle p_\perp^2\rangle$ to leading
order in energy in the opacity expansion. At leading order in $\alpha_s$ the
elastic energy loss gives a jet broadening that grows with $\ln E$. The
next-to-leading order in $\alpha_s$ result is a jet narrowing, due to
destructive LPM interference effects, that grows with $\ln^2 E$. We find that
in the opacity expansion the jet broadening asymptotics are -- unlike for the
mean energy loss -- extremely sensitive to the correct treatment of the finite
kinematics of the problem; integrating over all emitted gluon transverse
momenta leads to a prediction of jet broadening rather than narrowing. We
compare the asymptotics from the opacity expansion to a recent twist-4
derivation of $\langle p_\perp^2\rangle$ and find a qualitative disagreement:
the twist-4 derivation predicts a jet broadening rather than a narrowing.
Comparison with current jet measurements cannot distinguish between the
broadening or narrowing predictions. We comment on the origin of the difference
between the opacity expansion and twist-4 results.
