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The European Strategy for Particle Physics (ESSP) submitted in 2013 adeliberation document to the CERN council explaining that a lepton colliderwith "energies of 500\,GeV or higher could explore the Higgs propertiesfurther, for example the [Yukawa] coupling to the top quark, the [trilinear]self-coupling and the total width.". In view of the forthcoming ESPP update in2020, variations on this qualitative theme have been applied, inaccurately, tothe case of the ILC, to argue that an upgrade to 500\,GeV would allow themeasurement of the Higgs potential and would increase the potential for newparticle searches. As a consequence, the strategic question was raised againwhether the FCC-ee design study ought to consider a 500 GeV energy upgrade. Inthis note, we revisit the ESSP 2013 statement quantitatively and find [...]that 500 GeV is not a particularly useful energy for the lepton colliders underconsideration, especially for the FCC-ee. A 5 sigma demonstration of theexistence of the Higgs self-coupling is within reach at the energies foreseenfor the FCC-ee, with a moderate change of configuration, which certainlydeserves consideration.

Future strategies for the discovery and the precise measurement of the Higgs self coupling