Braverman–Makarychev–Makarychev–Naor conjecture on an explicit Krivine rounding scheme

Let $h_5$ be the fifth-degree Hermite polynomial, and for $\eta>0$ define functions on $\mathbb{R}^2$ by

$$f_\eta(x)=g_\eta(x)=\operatorname{sign}(x_2-\eta h_5(x_1)).$$

Braverman–Makarychev–Makarychev–Naor conjecture. There exists $\eta>0$ such that $f_\eta$ and $g_\eta$ form a Krivine rounding scheme and

$$c(f_\eta,f_\eta)>\frac{2}{\pi}\log(1+\sqrt{2}).$$

Equivalently, Grothendieck's inequality can be proved with the bound in the source without the auxiliary randomization step and the parameter $0<p<1$. The conjecture proposes an explicit Krivine rounding scheme that would remove the extra randomization used in the cited proof.