For a family of abelian varieties, the rational cohomology/relative Chow motive has a canonical decomposition into pure weight spaces by Beauville and Deninger-Murre. A degenerate abelian fibration is a fibration which is an abelian fibration generically but admits singular fibers on the boundary. It is a natural question to ask whether some of the structure remains to hold for degenerate abelian fibrations. In this talk, we consider perverse filtration on the rational cohomology of the relative compactified Jacobian and ask when the filtration has Fourier-stable multiplicative splitting. I will explain that if the relative compactified Jacobian arises from the Beauville-Mukai system, we get such multiplicative splitting. On the other hand, for the general family of integral curves, even for the family of nodal curves, we show that such multiplicative splitting cannot exist. This is a joint work with D. Maulik, J. Shen and Q. Yin.