Higher-dimensional density conjecture for Hessian-Schatten variation

Let $n\geq 2$, let $\Omega=(0,1)^n$, and consider functions with bounded Hessian--Schatten variation, together with the class $\operatorname{CPWL}(\Omega)$ of continuous piecewise-linear functions. Higher-dimensional density conjecture. The density result of Theorem 1 remains valid when the input domain is chosen to be any $n$-dimensional hypercube, namely, $\Omega=(0,1)^n$. Thus, in the corresponding higher-dimensional setting, $\operatorname{CPWL}(\Omega)$ functions should be dense in the Hessian--Schatten energy with respect to the topology and energy convergence asserted in the theorem. The two-dimensional theorem establishes this density result for $n=2$ and Schatten parameter $p=1$; the source reports that Sergio Conti announced a proof of the conjecture, so the proposed extension is considered resolved.