Some more Fano threefolds with a multiplicative Chow-Künneth decomposition

We exhibit several families of Fano threefolds with a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of these threefolds injects into cohomology. As a by-product of the argument, we observe that double covers of projective spaces admit a multiplicative Chow-Künneth decomposition.