Optimality of the noise threshold for three-outcome simulation of two binary measurements

Let $CS^1_{2,2;(t,t)}$ denote the set of pairs of binary measurements with common noise parameter $t$, and let $CS^1_{2,2}$ denote the set of pairs of binary measurements. A factorization through $S_3$ means that the identity map from $CS^1_{2,2;(t,t)}$ to $CS^1_{2,2}$ can be post-processed through a three-outcome measurement.

Optimality conjecture. The value $t\leq 1/3$ in the preceding result is optimal: pairs of noisy binary measurements with noise parameter $t>1/3$ cannot, in general, be simulated by post-processing a three-outcome measurement.

The preceding proposition establishes factorization through $S_3$ for $t\leq 1/3$; the conjecture asserts that this threshold cannot be improved.