  The observed scatter in intergalactic Lyman-$\alpha$ opacity at $z \lesssim
6$ requires large-scale fluctuations in the neutral fraction of the
intergalactic medium (IGM) after the expected end of reionization.
Post-reionization models that explain this scatter invoke fluctuations in
either the ionizing ultraviolet background (UVB) or IGM temperature. These
models make very different predictions, however, for the relationship between
Lyman-$\alpha$ opacity and local density. Here we test these models using
Lyman-$\alpha$ emitting galaxies (LAEs) to trace the density field surrounding
the longest and most opaque known Lyman-$\alpha$ trough at $z < 6$. Using deep
Subaru Hyper Suprime-Cam narrow-band imaging, we find a highly significant
deficit of $z \simeq 5.7$ LAEs within 20 Mpc/$h$ of the trough. The results are
consistent with a model in which the scatter in Lyman-$\alpha$ opacity near $z
\sim 6$ is driven by large-scale UVB fluctuations, and disfavor a scenario in
which the scatter is primarily driven by variations in IGM temperature. UVB
fluctuations at this epoch present a boundary condition for reionization
models, and may help shed light on the nature of the ionizing sources.
