The lemniscate starlike class Bohr-radius conjecture

Let $\mathcal{SL}^{*}$ denote the class of lemniscate starlike functions, associated with $\psi(z)=\sqrt{1+z}$. The Bohr-radius conjecture. The Bohr radius of $\mathcal{SL}^{*}$ is $r_0\approx0.439229$, where $r_0$ is the unique root in $(0,1)$ of

$$e^2 r \exp(2\sqrt{1+r}-2)=(1+\sqrt{1+r})^2.$$

This conjecture is proposed using the coefficient-bound conjecture for the lemniscate starlike class and the Bohr-phenomenon method described in the paper; the asserted uniqueness and numerical value remain unsupported by a resolution in the supplied text.