Stability of sharp Fourier restriction to spheres

In dimensions $d \in \{3,4,5,6,7\}$, we prove that the constant functions on the unit sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$ maximize the weighted adjoint Fourier restriction inequality $$\left| \int_{\mathbb{R}^d} |\widehat{f\sigma}(x)|^4\,\big(1 + g(x)\big)\,d x\right|^{1/4} \leq {\bf C} \, \|f\|_{L^2(\mathbb{S}^{d-1})}\,,$$ where $\sigma$ is the surface measure on $\mathbb{S}^{d-1}$, for a suitable class of bounded perturbations $g:\mathbb{R}^d \to \mathbb{C}$. In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ($g = {\bf 0}$), this was established by Foschi ($d=3$) and by the first and third authors ($d \in \{4,5,6,7\}$) in 2015. Our methods also yield a new sharp adjoint restriction inequality on $\mathbb S^7\subset \mathbb R^8$.