Existence conjecture for generalized reflectors with finitely many point targets

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

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

Generalized-reflector existence conjecture. There exists a generalized reflector $R\in \mathcal{R}_1^{\mathscr{C}_U^\delta(z')}(T)$ such that $G_1(\{x_i\})=f_i$ for every $i\in [k]$.

This conjecture proposes existence of a generalized reflector distributing the prescribed total measure among finitely many point targets. The paper establishes existence under certain assumptions but leaves this unrestricted finite-target statement as a possible direction for further research.