Trudinger–Wang affine Bernstein conjecture

Let $u\colon\mathbb{R}^n\to\mathbb{R}$ be smooth and locally uniformly convex. Write

$$U=(U^{ij})=(\det D^2u)(D^2u)^{-1},$$

and

$$w=[\det D^2u]^{-\frac{n+1}{n+2}}.$$

Assume that $u$ satisfies

$$\sum_{i,j=1}^{n}U^{ij}w_{ij}=0.$$

Trudinger–Wang's conjecture. If $n\leq 9$, then $u$ is an elliptic quadratic polynomial. If $n\geq 10$, then there exists a smooth, locally uniformly convex solution of the equation which is not an elliptic quadratic polynomial. The conjecture describes the sharp dimension threshold for the affine Bernstein problem: the low-dimensional assertion is known in the stated setting, while the higher-dimensional existence assertion is supported by the singular counterexample discussed in the source; the combined formulation is treated here as unresolved.