The Hessian conjecture for formal Legendre transforms

Let $K$ be a field of characteristic $0$, let $\varphi\in K[x_1,\ldots,x_n]$ be a polynomial whose Hessian determinant is a nonzero constant, and let $\bar\varphi$ be its formal Legendre transform, defined from the formal inverse of $\nabla\varphi$. Hessian conjecture. Then $\bar\varphi$ is also a polynomial. The conjecture is known when $K=\mathbb{R}$ and the Hessian matrix of $\varphi$ is positive or negative definite somewhere, and is equivalent to the Jacobian conjecture; the general case therefore remains open.