Bohr-radius conjecture for cardioid starlike functions

Let $\mathcal{S}^{*}_{\wp}$ denote the class of cardioid starlike functions associated with $\psi(z)=1+ze^z$, and let $r_0$ denote the Bohr radius for this class. Bohr-radius conjecture. The Bohr radius is $r_0\approx0.349681$, where $r_0$ is the unique root in $(0,1)$ of

$$re^{e^r}=e^{1/e}.$$

This conjecture follows from the proposed coefficient bounds and the extremal function for the cardioid starlike class; the preceding discussion identifies $r_0$ as an upper bound obtained from the Bohr-phenomenon argument, but the source provides no resolution of the conjecture.