Meng's Hessian conjecture

Let $\Bbbk$ be a field of characteristic zero and let $\varphi\in\Bbbk[x_1,\dots,x_n]$. Write

$$\operatorname{Hess}\varphi=\left(\frac{\partial^2\varphi}{\partial x_i\partial x_j}\right).$$

If $\det\operatorname{Hess}\varphi\in\Bbbk^{\times}$, the gradient map has a formal inverse and the formal Legendre transform is defined by

$$\varphi^{L}(p)=\langle p,x(p)\rangle-\varphi(x(p)),$$

where $x(p)$ is the formal inverse of $p=\nabla\varphi(x)$. Meng's Hessian conjecture. If $\varphi\in\Bbbk[x_1,\dots,x_n]$ has $\det\operatorname{Hess}\varphi\in\Bbbk^{\times}$, then $\varphi^{L}$ is a polynomial.

This is closely related to the Jacobian conjecture through the gradient map and formal Legendre transformation. The paper's abstract states that the Hessian conjecture holds for $n\le3$, is false for $n\ge5$, and remains open only for $n=4$.