The smooth monomial-Hessian conjecture

Let $S=\mathbb C[x_1,\ldots,x_n]$ and let $g\in S$ be homogeneous of degree $d\geq 3$, with $n\geq 2$. Assume that its Hessian $H(g)$ is a monomial. Smooth monomial-Hessian conjecture. Then $g$ is smooth if and only if

$$H(g)=(x_1\cdots x_n)^{d-2}$$

and $g$ is a Fermat polynomial. This asks whether the monomial-Hessian condition forces every smooth example to be the Fermat case; the surrounding proposition establishes the claim under the stated hypotheses, while the text notes that removing the Hessian assumption requires a separate question.