The extremal entire-function conjecture for the Bernstein problem

For the Bernstein problem with $a=0$, let $F$ be the extremal entire function and write

$$F(z)=z^p+(-1)^{[p/2]}E\cos\phi(z).$$

Here $\phi$ is a conformal map, and write its boundary curve as

$$\gamma=\{u+iv=\phi(x):x\in\mathbb{R}\}.$$

Extremal entire-function conjecture. The function $F$ has the displayed form, where $\phi$ maps the upper half-plane onto the region in the upper half-plane above $\gamma$, satisfies

$$L\sin v(x)\sinh u(x)=x,\qquad x\in\mathbb{R},$$

and is normalized by $\phi(0)=0$ and $\phi(z)\sim z$ as $z\to\infty$. The conjecture gives an explicit conformal-geometric description of the extremal function in the Bernstein problem; the supplied text does not state whether it has been proved or remains open.