= Solution
Conventional RF cavities accelerate beams with controlled phase, energy spread and transverse <beam emittance>. Their material surfaces limit sustainable gradients to order $10$–$100\,\mathrm{MV\,m^{-1}}$, depending on technology and operating conditions. Mature RF systems already reached multi-GeV and much higher beam energies in 2015. The 2013 ILC baseline proposed $250\,\mathrm{GeV}$ electrons colliding with $250\,\mathrm{GeV}$ positrons, approximately $31.5\,\mathrm{MV\,m^{-1}}$ superconducting cavities and a roughly $31\,\mathrm{km}$ facility. These were design parameters, not an operating collider.https://arxiv.org/abs/1306.6327[2013 ILC Technical Design Report, Volume 1: baseline 500 GeV centre-of-mass design]{ref}
In <plasma wakefield acceleration>, an intense laser displaces plasma electrons and excites a longitudinal charge-density wave. A suitably injected electron bunch gains energy from that wave, with fields of order tens to hundreds of $\mathrm{GV\,m^{-1}}$. A concrete result available before this examination was $4.2\,\mathrm{GeV}$ from a $9\,\mathrm{cm}$ capillary, with $6\%$ rms energy spread, $6\,\mathrm{pC}$ bunch charge and $0.3\,\mathrm{mrad}$ rms divergence. These values characterize that experiment, not every laser-plasma beam; divergence alone is not an emittance measurement.https://physics.aps.org/featured-article-pdf/10.1103/PhysRevLett.113.245002[2014 experiment, abstract and first page: measured energy, spread, charge, divergence and capillary length]{ref}
A collider concept stages many separately driven plasma modules, with fresh laser pulses, interstage capture and refocusing, controlled injection, and a final collision focus. Dephasing and pump depletion limit a single stage. Although the accelerating sections could be much shorter than ILC linacs, a useful collider also requires high repetition rate and wall-plug efficiency, reproducible low energy spread, preserved normalized emittance, stable timing, high bunch charge and acceptable beam-beam effects. Positron acceleration is an additional challenge because an electron-optimized plasma wake need not focus positrons. Large gradient alone does not demonstrate the luminosity or beam quality of an RF collider.
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