Exponentials rarely maximize Fourier extension inequalities for cones

We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in ${\mathbb{R}}^{1+d}$. In the range for which such inequalities are conjectural, our result is conditional on the boundedness of the extension operator. Global maximizers for the $L^2$ Fourier extension inequality on the cone in ${\mathbb{R}}^{1+d}$ have been characterized in the lowest-dimensional cases $d\in \{2,3\}$. We further prove that these functions are critical points for the $L^p$ to $L^q$ Fourier extension inequality if and only if $p=2$.