Conjecture on maximizers for the wave Strichartz inequality

For an integer $n\geq 2$, set $p=2+4/(n-1)$. Let $f_*$ be the function on $\mathbb{R}^n$ whose Fourier transform is

$$\widehat{f_*}(\xi)=|\xi|^{-1}\exp\bigl(-|\xi|\bigr),$$

and let $u_*$ be the solution of the wave equation with initial data

$$u_*(0)=f_*,\qquad \partial_tu_*(0)=0.$$

Wave maximizer conjecture. The set of maximizers for which equality holds in the sharp wave Strichartz inequality coincides with the set of initial data of solutions in the orbit of $u_*$ under the action of the group ${\mathcal L}$. This is the conjectural higher-dimensional analogue of the geometric characterization proved in the paper for the cases $(n,p)=(2,6)$ and $(3,4)$; the claim is stated for exponents that need not be even integers.