Support theorem of universal compactified Jacobians

We prove a full support theorem for the relative good moduli space of the universal compactified Jacobian $\bar{\pi}\colon \overline{J}_{g,n}^{d,\phi}\to \overline{\mathcal{M}}_{g,n}$, showing that every direct summand appearing in the BBDG decomposition of $\mathrm{R}\bar{\pi}_*\mathrm{IC}(\overline{J}_{g,n}^{d,\phi})$ has full support on the base $\overline{\mathcal{M}}_{g,n}$. Moreover, we explicitly describe this decomposition governed by the derived pushforward of the constant sheaf on the universal curve. The first proof synthesizes Maulik and Shen's generalization of Ng\^{o}'s support theorem, a decomposition theorem for the good moduli space morphism, and equivariant perverse sheaves. We also provide an independent second proof by variation of stability conditions and the support theorem for relative Jacobians by Migliorini, Shende, and Viviani.