Electron-hole recollisions are induced by resonantly injecting excitons with a near-IR laser at frequency $f_$ into quantum wells driven by a ~10 kV/cm field oscillating at $f_= 0.57$ THz. At $T=12$ K, up to 18 sidebands are observed at frequencies $f_=f_+2n f_$, with $-8 \le 2n \le 28$. Electrons and holes recollide with total kinetic energies up to 57 meV, well above the $E_} = 36$ meV threshold for longitudinal optical (LO) phonon emission. Sidebands with order up to $2n=22$ persist up to room temperature. A simple model shows that LO phonon scattering suppresses but does not eliminate sidebands associated with kinetic energies above $E_
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