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Let $(M,h)$ be a general polarized HK fourfold of Kummer type such that $q_M(h)\\equiv -6\\pmod{16}$ and the divisibility of $h$ is $2$, or $q_M(h)\\equiv -6\\pmod{144}$ and the divisibility of $h$ is $6$. We show that there exists a unique (up to isomorphism) slope stable vector bundle $\\cal F$ on $M$ such that $r({\\cal F})=4$, $ c_1({\\cal F})=h$, $\\Delta({\\cal F})=c_2(M)$. Moreover $\\cal F$ is rigid. 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