The conjecture that the planar rounding scheme suffices

Let $f_\eta:\mathbb R^2\to\{-1,1\}$ be the family of measurable functions defined in the paper, and let $c(f,g)$ denote the associated Krivine rounding constant. The planar rounding-scheme conjecture. For sufficiently small $\eta\in(0,1)$, the pair $f=g=f_\eta$ is a Krivine rounding scheme and

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

Equivalently, the proof should require only $H_\eta$, rather than a convex combination of $H_0$ and $H_\eta$. The conjecture is presented as a possible simplification of the paper's proof; no resolution is supplied.