In this paper, we show that it is NP-hard to determine whether a given graph admits a min-1-planar drawing. A drawing of a graph is min-$k$-planar if, for every crossing in the drawing, at least one of the two crossing edges involves at most $k$ crossings. This notion of min-$k$-planarity was introduced by Binucci, Büngener, Di Battista, Didimo, Dujmović, Hong, Kaufmann, Liotta, Morin, and Tappini [GD 2023; JGAA, 2024] as a generalization of $k$-planarity.

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