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More precisely, let $u \\ge 0$ be an integer and $A(n)$ be a random $n \\times (n+u)$ matrix over $\\mathbb{Z}_p$ whose $i$-th column is $\\alpha_n(i)$-balanced. We prove that if $\\sum_{i=1}^{n+u} \\exp(-\\epsilon \\alpha_n(i)n) \\to 0$ as $n \\to \\infty$ for every $\\epsilon>0$, then the cokernels of $A(n)$ converge in distribution, as $n \\to \\infty$, to the same limiting law as the cokernels of Haar-random $n \\times (n+u)$ matrices over $\\mathbb{Z}_p$. 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