Higher-dimensional polyhedral singularity conjecture for the Monge–Ampère equation

Let $\Omega \subset \mathbb{R}^n$ be a compact convex polytope, and let $\Gamma_k$ denote its $k$-skeleton. A convex function is a function whose epigraph is convex. Higher-dimensional polyhedral singularity conjecture. There exists a convex function $u: \mathbb{R}^n \rightarrow \mathbb{R}$ such that

$$\Gamma_{\left\lceil \frac{n}{2} - 1\right\rceil} \subset \{u = 0\}, \quad u \in C^{\infty}\left(\mathbb{R}^n \backslash\Gamma_{\left\lceil \frac{n}{2} - 1\right\rceil}\right), \quad \text{and} \quad \det D^2u = 1 + \sum_{q \in \Gamma_0} a_q\delta_q$$

for some coefficients $a_q > 0$. The conjecture would extend the paper's construction of solutions with polyhedral singularities from dimensions $n \leq 4$ to higher dimensions, provided the corresponding qualitative regularity result for the Monge–Ampère obstacle problem holds in those dimensions.