Foschi's Gaussian maximizer conjecture for the paraboloid extension inequality

Let $d\geq 3$. The quantity $\mathcal S_d$ denotes the supremum in the relevant paraboloid extension inequality over functions $\psi$ on $\mathbb{R}^d$. Foschi's Gaussian maximizer conjecture. The supremum defining $\mathcal S_d$ is attained for

$$\psi(x)=e^{-x^2/2},\qquad x\in\mathbb{R}^d.$$

This conjecture concerns the extremizers for the sharp paraboloid extension inequality. The cited work establishes the corresponding cases $d=1$ and $d=2$, while the assertion for all $d\geq 3$ is presented here as unproved.