  The purpose of this paper is to study the effect of conformal perturbations
on the local smoothing effect for the Schr\"odinger equation on surfaces of
revolution. The paper \cite{ChWu-lsm} studied the Schr\"odinger equation on
surfaces of revolution with one trapped orbit. The dynamics near this trapping
were unstable, but degenerately so. Beginning from the metric $g$ from this
paper, we consider the perturbed metric $g_s = e^{sf}g$, where $f$ is a smooth,
compactly supported function. If $s$ is small enough and finitely many
derivatives of $f$ satisfy appropriate symbolic estimates, then we show that a
local smoothing estimate still holds.
