Rotational symmetry conjecture for entire critical k-Hessian solutions

Let $u$ be a solution on $\mathbb{R}^n$ of the equation

$$\begin{cases} \sigma_k(-\nabla^2u)=e^{nu},\\ \sigma_1(-\nabla^2u),\ldots,\sigma_k(-\nabla^2u)>0, \end{cases}$$

with $\rho=0$ and $n=2k$. Here, $\sigma_j$ denotes the $j$-th elementary symmetric function of the eigenvalues of $-\nabla^2u$.

Rotational symmetry conjecture. Every such solution $u$ on $\mathbb{R}^n$ is rotationally symmetric.

This is a conjecture for critical-dimension $k$-Hessian equations, motivated by the paper's Liouville theorem and presented as a direction for future work. The source does not state a resolution, so its status remains open.