Trudinger–Wang conjecture on the affine mean-convex first boundary value problem

Let $\Omega\subset\mathbb{R}^n$ be the domain and let $\phi$ be the prescribed smooth, strictly convex boundary function in the first boundary value problem. Let $H_{\mathcal{A}}[\phi]$ denote the affine mean curvature of the graph of $\phi$, and call $\phi$ affine mean convex when

$$H_{\mathcal{A}}[\phi]>0.$$

Trudinger–Wang's conjecture. The first boundary value problem has a unique smooth solution $u$ if $\phi$ is affine mean convex. This conjecture concerns existence and uniqueness for the prescribed affine mean curvature equation with first boundary data. The source presents it as an open problem, following partial results for weak solutions and interior regularity.