The topological two-contact-point conjecture for cheese domains

Let $\Omega_0$ be a compact convex subset of $\mathbb{R}^2$ with non-empty interior. Let $\Omega_1$ be a convex open set such that

$$\operatorname{cl}\Omega_1 \subset \Omega_0.$$

Set $\Omega=\Omega_0\setminus\Omega_1$. For a bounded function $f\colon\partial_{\mathrm{fixed}}\Omega\to\mathbb{R}$, let $\mathfrak{B}_{\Omega,f}$ and $\mathfrak{B}_{\Omega_0,f}$ denote the corresponding Bellman functions. The two-contact-point conjecture. There are at least two points $x_1,x_2\in\partial_{\mathrm{free}}\Omega$ such that

$$\mathfrak{B}_{\Omega,f}(x_i)=\mathfrak{B}_{\Omega_0,f}(x_i),\qquad i=1,2.$$

This conjecture is motivated by the topology of a convex domain with a convex hole: optimizing martingales should not be able to turn over the inner boundary. The source presents it as the mechanism behind the extension of the main theorem to such domains; no resolution is supplied.