Some new Fano varieties with a multiplicative Chow-K\"unneth decomposition

Let $Y$ be a smooth dimensionally transverse intersection of the Grassmannian $\text{Gr}(2,n)$ with 3 Pl"ucker hyperplanes. We show that $Y$ admits a multiplicative Chow-K"unneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.