Ciliberto–Ottaviani conjecture on the birationality of Hessian maps

Let $S(n)$ be the graded polynomial ring in $x_0,x_1,\ldots,x_n$ with complex coefficients. For a degree $d$ homogeneous polynomial $f\in S(n)_d$, define its Hessian polynomial by

$$\operatorname{hess}(f)=\det\left(\frac{\partial^2f}{\partial x_i\partial x_j}\right).$$

For $d\geq 3$ and $n\geq 2$, this gives the rational Hessian map

$$h_{d,n}:\mathbb{P}(S(n)_d)\dashrightarrow\mathbb{P}(S(n)_{3(d-2)}),\qquad h_{d,n}(f)=\operatorname{hess}(f).$$

Ciliberto–Ottaviani conjecture. The map $h_{d,n}$ induces a birational morphism onto its image when $d\geq 3$, $n\geq 2$, and $(d,n)\neq(3,2)$. This conjecture concerns the birationality of Hessian maps for homogeneous polynomials. The binary case $n=1$ is known: the corresponding map is birational onto its image if and only if $d\geq 5$. The cubic-surface case is also known, while the stated claim includes quartic plane curves and excludes the exceptional pair $(3,2)$; the source presents it as an open conjecture.