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We prove the generalized Franchetta conjecture for the locally completefamily of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten(LLSS). As a corollary, we establish the Beauville-Voisin conjecture for verygeneral LLSS eightfolds. The strategy consists in reducing to the Franchettaproperty for relative fourth powers of cubic fourfolds, by using the recentdescription of LLSS eightfolds as moduli spaces of semistable objects in theKuznetsov component of the derived category of cubic fourfolds, together withits generalization to the relative setting due toBayer-Lahoz-Macr\`i-Nuer-Perry-Stellari. As a by-product, we compute the Chowmotive of the Fano variety of lines on a smooth cubic hypersurface in terms ofthe Chow motive of the cubic hypersurface.

The generalized Franchetta conjecture for some hyper-K\"ahler varieties, II