Conjecture on the distance formula and strong exposedness of the functions $f_\beta$

Let $A^1$ be the Bergman space, let $(A^1)^\perp$ denote its annihilator in the relevant dual space, let $C$ denote the subspace of constant functions, and let $\varphi_\beta$ and $f_\beta$ be the functions defined in the paper for $-2<\beta<0$. For a point $x$ and subset $E$ of a normed space, write $d(x,E)$ for the distance from $x$ to $E$. Distance and exposedness conjecture. For all $-2<\beta<0$,

$$d(\varphi_\beta,(A^1)^\perp+C)=\frac{2}{\pi}\frac{|\sin(\frac{\beta\pi}{2})|}{\beta+2}.$$

In particular, the functions $f_\beta$ are strongly exposed for all such $\beta$. This conjecture extends the established strong exposedness result for $-1<\beta<0$ and is motivated by the asymptotic sharpness of the distance inequality as $\beta$ decreases to $-2$; its resolution for the full interval remains open in the supplied text.