Polynomiality conjecture for Burgers' equations with nilpotent Hessian

Let $P(z)$ be a homogeneous polynomial of degree $d\geq 2$ whose Hessian matrix $\operatorname{Hes}(P)$ is nilpotent. Let $U_t(z)$ be the unique power series solution of the Cauchy problem

$$\frac{\partial U_t}{\partial t}(z)=\frac{1}{2}\langle\nabla U_t(z),\nabla U_t(z)\rangle, \qquad U_{t=0}(z)=P(z).$$

Polynomiality conjecture. The solution $U_t(z)$ must be a polynomial in both $z$ and $t$. This conjecture links the nilpotent-Hessian condition to polynomial solutions of the inviscid Burgers equation and is motivated by its relationship with the Jacobian conjecture. The supplied text does not state a resolution.