Binary-alphabet conjecture for the optimal auxiliary random variable

Let $U$ be the auxiliary random variable in the feedback-based collaborative secrecy encoding problem over binary symmetric channels. The preceding result establishes that $U$ can be restricted without loss of optimality to a three-dimensional alphabet. Binary-alphabet conjecture. The optimal auxiliary random variable $U$ can be defined, without loss of optimality, over a binary alphabet. If true, this would reduce the search for the boundary of the achievable common-message and secret-message rate region to binary auxiliary variables; the supplied text presents arguments supporting the conjecture but does not state a resolution.