The Kakimizu complex of a connected sum of links

We show that $|MS(L_1 # L_2)|=|MS(L_1)|\times|MS(L_2)|\times\mathbb{R}$ when $L_1$ and $L_2$ are any non-split and non-fibred links. Here $MS(L)$ denotes the Kakimizu complex of a link $L$, which records the taut Seifert surfaces for $L$. We also show that the analogous result holds if we study incompressible Seifert surfaces instead of taut ones.