Maximal non-Kochen–Specker set conjecture

Let a non-Kochen–Specker set in dimension $d$ be a measurable subset $A\subseteq S^{d-1}$ admitting a valuation map $v:A\to\{0,1\}$ such that $v(-n)=v(n)$, every mutually orthogonal collection has valuation sum at most $1$, and every set of $d$ mutually orthogonal directions has valuation sum $1$. In the construction discussed, $N=N_0\cup N_1\subseteq S^2$ is a non-Kochen–Specker set, with normalized measure $|N|=2-\sqrt{2}+\frac{\sqrt{2}\ln 2}{\pi}\approx0.8978$. Maximal non-Kochen–Specker set conjecture. $N$ is the non-Kochen–Specker set of largest measure. The preceding calculation establishes only the lower bound $|N_{\mathrm{max}}|\geq 2-\sqrt{2}+\frac{\sqrt{2}\ln 2}{\pi}\approx0.8978$; whether this particular set is globally maximal is not resolved in the supplied context.