Griffiths–Harris rigidity conjecture for the Segre variety

Let $X=\operatorname{Seg}(\mathbb P^2\times\mathbb P^2)\subset\mathbb P(\mathbb C^3\otimes\mathbb C^3)$. Let $Z^4\subset\mathbb P V$ be a variety, and let $II_{Z,z}$ denote its second fundamental form at a general point $z\in Z$. Griffiths–Harris conjecture. If, at $z\in Z_{general}$, $II_{Z,z}=II^X$, then $Z$ is projectively equivalent to the Segre variety $X$. This is a second-order rigidity assertion: the second fundamental form should characterize the Segre variety among four-dimensional varieties. The conjecture is presented as a question of Griffiths and Harris, with no resolution supplied here.