\begin{document}$M_{A^0}\simeq M_{H^0}>M_{H^\pm}>M_{H^{\pm\pm}}$\end{document}. In this study, we investigate this signature at future lepton colliders, such as the ILC, CLIC, and MuC. Direct searches for doubly charged scalar \begin{document}$H^{\pm\pm}$\end{document} at the LHC have excluded \begin{document}$M_{H^{\pm\pm}}<350(870)$\end{document} GeV in the \begin{document}$H^{\pm\pm}\to W^\pm W^\pm (\ell^\pm\ell^\pm)$\end{document} decay mode. Therefore, we choose \begin{document}$M_{A^0}=400,600,1000,1500$\end{document} GeV as our benchmark scenarios. Constrained by direct search, \begin{document}$H^{\pm\pm}\to W^\pm W^\pm$\end{document} is the only viable decay mode for \begin{document}$M_{A^0}=400$\end{document} GeV at the \begin{document}$\sqrt{s}=1$\end{document} TeV ILC. With an integrated luminosity \begin{document}$\mathcal{L}=8~ \mathrm{ab}^{-1}$\end{document}, the promising region, with approximately 150 signal events, corresponds to a narrow band in the range of \begin{document}$10^{-4}~\text{GeV}\lesssim v_\Delta \lesssim10^{-2}$\end{document} GeV. Meanwhile, for \begin{document}$M_{A^0}=600$\end{document} GeV at the \begin{document}$\sqrt{s}=1.5$\end{document} TeV CLIC, approximately 10 signal events can be produced with \begin{document}$\mathcal{L}=2.5~\text{ab}^{-1}$\end{document}. For heavier triplet scalars \begin{document}$M_{A^0}\gtrsim 870$\end{document} GeV, although the \begin{document}$H^{\pm\pm}\to \ell^\pm \ell^\pm$\end{document} decay mode is allowed, the cascade decays are suppressed. A maximum event number \begin{document}$\sim 16$\end{document} can be obtained at approximately \begin{document}$v_\Delta\sim4\times10^{-4}$\end{document} GeV and \begin{document}$\lambda_4\sim0.26$\end{document} for \begin{document}$M_{A^0}=1000$\end{document} GeV with \begin{document}$\mathcal{L}=5~ \mathrm{ab}^{-1}$\end{document} at the \begin{document}$\sqrt{s}=3$\end{document} TeV CLIC. Finally, we find that this signature is not promising for \begin{document}$M_{A^0}=1500$\end{document} GeV at the \begin{document}$\sqrt{s}=6$\end{document} TeV MuC. Based on the benchmark scenarios, we also study the observability of this signature. In the \begin{document}$H^{\pm\pm}\to W^\pm W^\pm(\ell^\pm\ell^\pm)$\end{document} mode, one can probe \begin{document}$M_{A^0}\lesssim800(1160)$\end{document} GeV at future lepton colliders."> Same-sign tetralepton signature in type-II seesaw at lepton colliders -
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