On extremizing sequences for the adjoint restriction inequality on the cone

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.