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We show that the closing probability $\\mathsf{W}_n \\big( \\vert\\vert \\Gamma_n \\vert\\vert = 1 \\big)$ that $\\Gamma$'s endpoint neighbours the origin is at most $n^{-1/2 + o(1)}$ in any dimension $d \\geq 2$. The method of proof is a reworking of that in [4], which found a closing probability upper bound of $n^{-1/4 + o(1)}$. 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We show that the closing probability $\\mathsf{W}_n \\big( \\vert\\vert \\Gamma_n \\vert\\vert = 1 \\big)$ that $\\Gamma$'s endpoint neighbours the origin is at most $n^{-1/2 + o(1)}$ in any dimension $d \\geq 2$. The method of proof is a reworking of that in [4], which found a closing probability upper bound of $n^{-1/4 + o(1)}$. 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