The dyadic Bellman-function concave-envelope conjecture

Let $Q\subset\mathbb{R}^d$ be a cube, let $\mathrm{BMO}_{\varepsilon}^{\mathrm{dyad}}(Q)$ denote the dyadic BMO class with parameter $\varepsilon$, and define

$$\boldsymbol{B}_{\varepsilon}^{\mathrm{dyad}}(x_1,x_2;f)=\sup\big\{f[\varphi]\mid \langle\varphi\rangle_Q=x_1,\ \langle\varphi^2\rangle_Q=x_2,\ \varphi\in\mathrm{BMO}_{\varepsilon}^{\mathrm{dyad}}(Q)\big\}.$$

An extension $\Omega$ of $\Omega_{\varepsilon}$ is a domain of the form

$$\Omega=\{x\in\mathbb{R}^2\mid x_2\geqslant x_1^2\}\setminus\Omega',$$

where $\Omega'$ is open, convex and unbounded, and its closure lies in $\{x\in\mathbb{R}^2\mid x_2>x_1^2+\varepsilon^2\}$. For every $f$, $\varepsilon$, and $d$, there exists an extension $\Omega$ of $\Omega_{\varepsilon}$ such that

$$\boldsymbol{B}_{\varepsilon}^{\mathrm{dyad}}(x)=\inf\Big\{G(x)\mid G\text{ is locally concave on }\Omega,\quad G(x_1,x_1^2)\geqslant f(x_1)\text{ for all }x_1\in\mathbb{R}\Big\},\quad x\in\Omega_{\varepsilon}.$$

This conjecture asserts that the dyadic Bellman function is the restriction of the least locally concave majorant determined by the boundary data on the parabola, after choosing a suitable extension of the domain. The paper notes that the dyadic Bellman function satisfies a stronger dyadic concavity inequality than ordinary local concavity; the claimed identification of these two constructions remains unresolved here.