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We measured the branching fractions of the decays ${\ensuremath{\chi}}_{cJ}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{-}}{\overline{\mathrm{\ensuremath{\Sigma}}}}^{+}$ for the first time using the final states $n\overline{n}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$. The data sample exploited here is $448.1\ifmmode\times\else\texttimes\fi{}{10}^{6}\text{ }\text{ }\ensuremath{\psi}(3686)$ events collected with BESIII. We find $\mathcal{B}({\ensuremath{\chi}}_{cJ}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{-}}{\overline{\mathrm{\ensuremath{\Sigma}}}}^{+})=(51.3\ifmmode\pm\else\textpm\fi{}2.4\ifmmode\pm\else\textpm\fi{}4.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5},(5.7\ifmmode\pm\else\textpm\fi{}1.4\ifmmode\pm\else\textpm\fi{}0.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, and $(4.4\ifmmode\pm\else\textpm\fi{}1.7\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, for $J=0$, 1, 2, respectively, where the first uncertainties are statistical and the second …