König's extremal-kernel conjecture for Grothendieck's inequality

For a measure-space formulation of Grothendieck's inequality, let $K:\mathbb R^n\times\mathbb R^n\to\mathbb R$ be the oscillatory Gaussian kernel

$$K(x,y)=\exp\left(-\frac{\|x\|_2^2+\|y\|_2^2}{2}\right)\sin(\langle x,y\rangle),$$

and let

$$B_K(f,g)=\int_{\mathbb R^n}\int_{\mathbb R^n}f(x)g(y)K(x,y)\,dx\,dy.$$

Define $f_0:\mathbb R^n\to\{-1,1\}$ by $f_0(x_1,\ldots,x_n)=\operatorname{sign}(x_1)$. König's conjecture. For every $n\in\mathbb N$ and every measurable $f,g:\mathbb R^n\to\{-1,1\}$,

$$B_K(f,g)\leqslant B_K(f_0,f_0).$$

This conjecture proposes that the coordinate-sign functions are extremal for this kernel; the paper's construction gives a counterexample to König's problem.