Nearest-neighbor optimality for the general one-dimensional classification setting

Let $x$, $y$, and $z$ be three points on the real line, where $x$ and $z$ are generated independently from two distinct distributions and $y$ is generated from their 50--50 mixture. The hypotheses and nearest-neighbor rule are those of Theorem 1 in the paper. Nearest-neighbor optimality conjecture. Under these hypotheses, the optimal rule for deciding whether $y$ came from the same distribution as $x$ or $z$ is the nearest-neighbor rule. The question is left open after the paper proves that the rule is better than chance in the stated setting but does not prove its optimality.