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In this paper, we prove that every $k$-iterated line digraph of minimum in-degree at least 2 and $k\\geq2$, or minimum in-degree at least 3 and $k\\geq1$, admits a $(1,\\le \\ell)$-identifying code with $\\ell\\leq2$, and in any case it does not admit a $(1,\\le \\ell)$-identifying code for $\\ell\\geq3$. 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