Approximation conjecture for solutions of the inner-function curvature equation

Let $I$ be an inner function on the unit disk, let $I'$ denote its derivative, and let $u_{I',\infty}$ be the upper envelope associated with the equation below. Consider solutions $u$ on the unit disk satisfying

$$\Delta u=|I'|^2e^{2u}.$$

Approximation conjecture. Any solution $u$ of this equation such that $u\leq u_{I',\infty}$ can be approximated uniformly on compact subsets by solutions $u_k$ that are bounded above. This PDE assertion is stated as the intuitively plausible ingredient from which the paper deduces its main conjecture. The supplied text does not establish the approximation claim or provide a resolution of it.