Incompressibility of orthogonal Grassmannians of rank 2

For a nondegenerate quadratic form φ on a vector space V of dimension 2n+1, let Xd be the variety of d-dimensional totally isotropic subspaces of V. We give a sufficient condition for X2 to be 2-incompressible, generalizing in a natural way the known sufficient conditions for X1 and Xn. Key ingredients in the proof include the Chernousov–Merkurjev method of motivic decomposition as well as Pragacz and Ratajskiʼs characterization of the Chow ring of (X2)E, where E is a field extension splitting φ.