The ring-domain automatic concavity conjecture

Let $S^1=\{y\in\mathbb{R}^2\mid y_1^2+y_2^2=1\}$, and let $\mathfrak{B}^\circ_\delta[f]$ and $\mathfrak{B}^{\circ,*}_\delta[f]$ denote, respectively, the minimal locally concave and subtangentially locally concave functions on the ring

$$\{y\in\mathbb{R}^2\mid 1-\delta\leqslant y_1^2+y_2^2\leqslant 1\},$$

with boundary data $f\colon S^1\to\mathbb{R}$. Ring-domain automatic concavity conjecture. If $f\colon S^1\to\mathbb{R}$ is continuous, then

$$\mathfrak{B}^\circ_\delta[f]=\mathfrak{B}^{\circ,*}_\delta[f]$$

for every $\delta>0$. This would extend the automatic concavity phenomenon from the standard domain to the annular setting, which is not covered by the available theory for differences of unbounded convex sets. The source gives no resolution.