  Progress on classifying small index subfactors has revealed an almost empty
landscape. In this paper we give some evidence that this desert continues up to
index 3+\sqrt{5}. There are two known quantum-group subfactors with index in
this interval, and we show that these subfactors are the only way to realize
the corresponding principal graphs. One of these subfactors is
1-supertransitive, and we demonstrate that it is the only 1-supertransitive
subfactor with index between 5 and 3+\sqrt{5}. Computer evidence shows that any
other subfactor in this interval would need to have rank at least 38. We prove
our uniqueness results by showing that there is a unique flat connection on
each graph. The result on 1-supertransitive subfactors is proved by an argument
using intermediate subfactors, running the `odometer' from the FusionAtlas`
Mathematica package and paying careful attention to dimensions.
