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We prove that the $2$-complex associated to the presentation $\langle a,b\mid b,bab^{-1}a^{-2}\rangle$ has contracting non-positive immersions andpositive maximal irreducible curvature. This example shows that the contractingnon-positive immersions property is not equivalent to the notion ofnon-positive irreducible curvature, answering a question raised by H. Wilton.

A $2$-complex with contracting non-positive immersions and positive maximal irreducible curvature