On the Chow ring of Fano varieties of type $S2$

We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito-Miura-Okawa-Ueda and by Fatighenti-Mongardi) have a multiplicative Chow-K"unneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.