Existence conjecture for convex weak refractors with general target sets

Let $T$ be a target set satisfying Hypothesis H1, and let

$$K=\frac{\max_{x\in T}|x|}{\min_{x\in T}|x|}.$$

Suppose that $\epsilon_0<\lim_{t\to K^+}\frac{1}{t-1}$, where $\epsilon_0$ is defined by the source's equation (e0). Given $\gamma>0$ with

$$\epsilon_0+\gamma<\lim_{t\to K^+}\frac{1}{t-1},$$

set $\delta_\gamma=1/(\epsilon_0+\gamma)$. Let $g\in L^1(\mathbb{S}^2)$ be nonnegative and vanish outside $D_T^{\delta_\gamma}$, and let $f\in L^1(T)$ be nonnegative. If $F$ is the measure defined by the source's equation (lebesgue) and

$$F(T)=\mu_g(D_T^{\delta_\gamma}),$$

then there exists $\epsilon_M\in(\epsilon_0+\gamma,\lim_{t\to K^+}1/(t-1))$ and a convex refractor $R\in\mathcal{R}_{\mathrm{convex}}^{\epsilon_M}(T)$ that is a convex weak solution to the refractor problem.

Convex refractor existence conjecture. Under these hypotheses, such a convex weak solution exists for the target set and data described above.

The conjecture would extend the paper's existence results beyond the rotationally symmetric and sufficiently close target-set cases, potentially covering general exotic Lebesgue-measurable target sets. The source provides indirect evidence from its established existence theorems, but does not state that this general claim has been proved.