Convexity conjecture for illumination bodies in nonpositive-curvature space forms

Let $n\geq 3$, let $\lambda\leq 0$, and let $K\in\mathcal{K}_0(\mathrm{Sp}^n(\lambda))$ be a convex body. Illumination-body convexity conjecture. For every $\delta>0$, the illumination body satisfies

$$\mathcal{I}_\delta^\lambda(K)\in\mathcal{K}_0(\mathrm{Sp}^n(\lambda)).$$

The claim extends the proven two-dimensional hyperbolic result to higher-dimensional space forms of nonpositive curvature. In Euclidean space illumination bodies are convex, while the paper notes that convexity can fail in positive curvature; the higher-dimensional nonpositive-curvature case remains open.