Future prospects of mass-degenerate Higgs bosons in the $CP$-conserving two-Higgs-doublet model

The scenario of two mass-degenerate Higgs bosons within the general two-Higgs-doublet model (2HDM) is revisited. We focus on the global picture when two $CP$-even Higgs bosons of $h$ and $H$ are nearly mass-degenerate. A global fit to the signal strength of the 125 GeV Higgs measured at the LHC is performed. Based on the best-fit result of the 2HDM mixing angles $(\alpha ,\beta )$, theoretical constraints, charged and $CP$-odd Higgs boson direct search constraints and the electroweak precision constraints are imposed to the 2HDM parameter space. We present the signal predictions of the $(4b,, 2b,2\gamma)$ channels for the benchmark models at the LHC 14 TeV runs. We also study the direct Higgs boson pair productions at the LHC, and the Z-associated Higgs boson pair production search at the ILC 500 GeV runs, as well as the indirect probes at the CEPC 250 GeV run. We find that the mass-degenerate Higgs boson scenario in the Type-II 2HDM can be fully probed by these future experimental searches.