is a “BCS-like”.

However it has been found— unexpectedly for many researchers— that the BCS theory and its extension [@eli] towards the intermediate coupling regime, $\lambda \lesssim
 1$, break down already at $\lambda \gtrsim 1$ [@ale0]. It happens since the Migdal “noncrossing” approximation [@mig] of the theory is not applied at $\lambda \gtrsim1$. In fact, the small parameter of the theory, $\lambda\omega_0/E_F$, becomes large at $\lambda \gtrsim 1$ because the bandwidth is narrowed and the Fermi energy, $E_F$ is renormalised down exponentially due to the small polaron formation [@ale0; @alexandrov:2001] (here $\omega_0$ is the characteristic phonon frequency, and we take $\hbar=c=k_B=1$). Extending the BCS theory towards the strong interaction between electrons and ion vibrations, $\lambda \gg 1$, a charged Bose gas of tightly bound small bipolarons was predicted [@aleran] instead of Cooper pairs, with a further prediction that the highest superconducting transition temperature is attained in the crossover region of EPI strength, $\lambda \thickapprox 1$, between the BCS and bipolaronic superconductivity [@ale0].

For a very strong EPI polarons become self-trapped on a single lattice site and bipolarons are on-site singlets. A finite on-site bipolaron mass appears only in the second order of polaron hopping, [@aleran], so that on-site bipolarons might 