Conjecture on the Bohr radius for bounded functions with boundary restriction

Let $R_{id_1\rightarrow \partial}(a)$ denote the Bohr radius in the problem under consideration with the restriction $a>r$. Let $a_1=0.321037\ldots$ be the root of

$$1-\sqrt{\dfrac{1+a}{1+2a}}=a^2.$$

Bohr-radius conjecture. If $a\in(a_1,1]$, then

$$R_{id_1\rightarrow \partial}(a)=\dfrac{1}{a}\Big(1-\sqrt{\dfrac{1+a}{1+2a}}\Big).$$

This conjecture asserts that, when the only restriction is $a>r$, the Bohr radius reaches the known upper bound. The supplied text gives no resolution status.