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No evidence of this decay mode is found, with $\\mathcal{B}[J/\\psi\\rightarrow\\gamma X(2370)] \\times \\mathcal{B}[ X(2370) \\rightarrow K^{*}(892)^{0}\\bar{K}^{0}+\\mathrm{c.c.} \\rightarrow K^{0}_{S}K^{0}_{S}\\pi^{0}] < 2.7\\times 10^{-6}$ at the $90\\%$ confidence level. The suppression of the $K^{*}(892)\\bar{K}$ mode indicates the $X(2370)$ as a flavor-singlet state. 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