Higgs self-coupling measurement at future $e^+e^-$ colliders

The Higgs mechanism is a central part of the Standard Model which has not yet been fully established experimentally without the measurement of the Higgs self-coupling. Future linear $e^+e^-$ colliders are able to access centre-of-mass energies of 500 GeV and beyond and can therefore probe the Higgs self-coupling directly through the measurement of double Higgs production. A new analysis of the capability to measure the double Higgs-strahlung, $e^+e^-\to ZHH$, at a centre-of-mass energy of 500 GeV is ongoing based on the detailed, Geant4-based simulation of the ILD detector concept. This study has identified several aspects concerning the reconstruction techniques to fully exploit the detector potential, which are expected to improve precision reach and will be presented in this contribution. Additionally, the requirements that the Higgs self-coupling measurement puts on the choice of centre-of-mass energy will be evaluated as this is important for shaping the landscape of future colliders such as ILC or $C^3$.