What pairs electrons in doped SrTiO3 at Fermi energies below phonons

Electron-doped SrTiO3 (Nb, La or oxygen vacancies) superconducts from $n\ \text{below}\ 1\mathrm{e}18\,\mathrm{cm}^{-3}\ \text{to about}\ 1\mathrm{e}21\,\mathrm{cm}^{-3}$ with maximal $T_c$ of approximately $0.4\,\mathrm{K}$ near $n \sim 1\mathrm{e}20\,\mathrm{cm}^{-3}$; at the lowest densities $E_F \sim 1\,\mathrm{meV}$ is far below the longitudinal optical phonon energies (up to about $100\,\mathrm{meV}$). Identify the pairing interaction (soft transverse-optical mode via two-phonon or Rashba-type coupling, dynamically screened longitudinal-optical phonons and plasmons, or other) that reproduces $T_c(n)$ quantitatively over the full dome without adjustable couplings. An answer is a mechanism with computed $T_c(n)$ within a factor of about 2.

A first-principles strong-coupling calculation reproducing $T_{c}(n)$, the isotope effect and the gap-to-$T_{c}$ ratio across the dome.