Gaussian maximizer conjecture for paraboloid Strichartz inequalities

Let $d\geq 1$, let

$$e^{it\Delta}f(x):=\int_{\mathbb{R}^d}e^{2\pi i x\xi-4\pi^2it|\xi|^2}\widehat{f}(\xi)\,d\xi,$$

and set $q=2+4/d$. The paraboloid Strichartz inequality is

$$\|e^{it\Delta}f\|_{L^q(\mathbb{R}^{d+1})}\leq \mathbf{S}_d\|f\|_{L^2(\mathbb{R}^d)}.$$

Define

$$\mathcal{G}:=\left\{g:g=e^{it_0\Delta}\left[\lambda^{d/2}e^{2\pi i(x-x_0)\xi_0}e^{-\pi|\lambda(x-x_0)|^2}\right],\;(\lambda_0,\xi_0,x_0,t_0)\in\mathbb{R}_{+}\times\mathbb{R}^3\right\}.$$

Gaussian maximizer conjecture. The maximizers for the paraboloid Strichartz inequality are exactly the Gaussian-type functions $g\in\mathcal{G}$. This conjecture has been confirmed for $d=1,2$ and remains open for $d\geq 3$.