The real-analytic boundary conjecture for martingale connectedness

Let $w\subset\mathbb{R}^d$ be a bounded closed connected set with non-empty interior, and suppose that its boundary is a real-analytic submanifold of $\mathbb{R}^d$. The real-analytic boundary conjecture. Then $w$ is martingale connected. The conjecture would identify a broad geometric condition ensuring the existence of martingales connecting points through the domain. The source contrasts it with examples where the minimal principle fails and states no proof or resolution.