Liouville conjecture for entire 2-convex solutions of the 2-Hessian equation

Let $u\in C^4(\mathbb{R}^n)$ be an entire 2-convex function, meaning that its Hessian eigenvalues lie in the closure of the $2$-Hessian ellipticity cone, and suppose that

$$\sigma_2(D^2u(x))=1$$

throughout $\mathbb{R}^n$. Assume also that $u$ has the quadratic growth specified in the source. Liouville conjecture. Every such solution is a quadratic polynomial. This conjecture asks whether the lower bound on $\sigma_3(D^2u)$ in the preceding theorem can be removed; the paper proves the corresponding result under that bound, and proves it without the bound when $n=3$.