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Both the $\epem$ Higgs factories discussed in Section~\ref{sec:Higgs} are accompanied by high-energy upgrade projects. For the ILC, it is foreseen to double its length to reach a centre-of-mass energy of 500 GeV, and it is not yet excluded to extend it by another factor of two towards $\sqrt{s} =$ 1 TeV. Another linear collider project, CLIC~\cite{cite:CLICDR}, could also take over for the high-energy physics programme all the way to $\sqrt{s}=$ 3 TeV. The foreseen upgrade of TLEP is of another, unique, nature. It would consist in re-using the 80-to-100 km tunnel for a very-large-energy large hadron collider (VHE-LHC). If instrumented with magnets of 16 T, pp collisions would be produced at a centre-of-mass energy of 80 to 100 TeV.   Typically, $\epem$ colliders can pair-produce new particles with masses up to half the centre-of-mass energy, should they be either electrically charged or with a non-vanishing coupling to the Z. The reach of ILC500, ILC1000 and CLIC is therefore limited to particles lighter than 250, 500 and 1500 GeV/$c^2$, respectively. With the absence of new phenomena discovery at the LHC so far, air is therefore getting extremely thin for the ILC energy-frontier upgrade, even in its hypothetical 1-TeV version. The next LHC run at 13 TeV, expected to start in 2015, will bring the final word in this respect. A discovery of a new particle lighter than 1.5 TeV/$c^2$ in the 13-TeV LHC data would probably rejuvenate the proposal of CLIC at $\sqrt{s} =$ 3 TeV. Neutral particles with sizable direct coupling to fermions -- like new gauge bosons, for example -- can also be produced at $\epem$ colliders as $s$-channel resonances all the way to the highest centre-of-energy, but those are already excluded up to several TeV/$c^2$ by the LHC data at 8 TeV, and have therefore reduced interest in this context. A 100 TeV proton-proton collider would instead be able to produce new particles up to several tens of TeV/$c^2$, thus opening a unique window at high energy. A detailed study of the VHE-LHC physics case is therefore in order, and is foreseen to start at the beginning of 2014. 2014, in order to have relevant answers ready for the next European Strategy update to take an informed decision in 2018.