Existence conjecture for interpolated reflectors with finitely many point targets

Let $U$ be a simply connected open set in $\mathbb{S}_+^2$, let $\delta,z'>0$, and let $\{x_1,\dots,x_k\}\in \mathbb{R}^{3-}$. Suppose that $g\in L^1(\mathbb{S}^2)$ is nonnegative and satisfies $g\equiv 0$ outside $U$. Let $f_1,\dots,f_k$ be nonnegative real numbers satisfying

$$\sum_{i=1}^k f_i=\mu_g(U).$$

Interpolated-reflector existence conjecture. There exists an interpolated reflector $R\in \mathcal{R}_2^{\mathscr{C}_U^\delta(z')}(T)$ such that $G_2(\{x_i\})=f_i$ for every $i\in [k]$.

This conjecture would provide interpolated reflector designs realizing arbitrary prescribed masses at finitely many point targets. The paper notes that such designs would be practically useful, while the stated existence claim remains open.