  We prove the generalized Franchetta conjecture for the locally complete
family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten
(LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very
general LLSS eightfolds. The strategy consists in reducing to the Franchetta
property for relative fourth powers of cubic fourfolds, by using the recent
description of LLSS eightfolds as moduli spaces of semistable objects in the
Kuznetsov component of the derived category of cubic fourfolds, together with
its generalization to the relative setting due to
Bayer-Lahoz-Macr\`i-Nuer-Perry-Stellari. As a by-product, we compute the Chow
motive of the Fano variety of lines on a smooth cubic hypersurface in terms of
the Chow motive of the cubic hypersurface.
