  This small note is about Pl\"ucker hyperplane sections $X$ of the
Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic
fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by
Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also
prove that a certain subring of the Chow ring of $X$ (containing all
intersections of positive-codimensional subvarieties) injects into cohomology.
