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The polygon cardinality grows exponentially, and the growth rate $\\lim_{n \\in 2\\mathbb{N}} p_n^{1/n} \\in (0,\\infty)$ is called the connective constant and denoted by $\\mu$. Madras [J. Statist. Phys. 78 (1995) no. 3--4, 681--699] has shown that $p_n \\mu^{-n} \\leq C n^{-1/2}$ in dimension $d=2$. Here we establish that $p_n \\mu^{-n} \\leq n^{-3/2 + o(1)}$ for a set of even $n$ of full density when $d=2$. 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