  It is known that extremizers for the $L^2$ to $L^6$ adjoint Fourier
restriction inequality on the cone in $\mathbb{R}^3$ exist. Here we show that
nonnegative extremizing sequences are precompact, after the application of
symmetries of the cone. If we use the knowledge of the exact form of the
extremizers, as found by Carneiro, then we can show that nonnegative
extremizing sequences converge, after the application of symmetries.
