Outer-function characterization in the upper half-plane

Let $(G,\theta)$ be a compactification, let $h\in H^p(G,\theta)$ be a continuous function with bounded spectrum contained in $[0,\infty)$, and let $f=h\circ\theta$. Then $f$ extends to an entire function $F$ bounded in the upper half-plane. The function $h$ is outer when it satisfies the outer-function integral condition

$$\int_G\log|h|=\log\left|\int_Gh\right|.$$

Outer-function characterization. The function $h$ above is outer if and only if $F$ has no zeros in the open upper half-plane.

This connects Hardy-space factorization on the compactification with zero-freeness of the associated entire function. The excerpt presents the assertion without indicating whether it is proved or remains open.