bipolarons were found in 1D HHM using variational methods also in the non and near-adiabatic regimes with dynamical quantum phonons [@lamagna; @trugman]. The intersite bipolaron with a relatively small effective mass is stable in a wide region of the parameters of HHM due to both exchange and nonadiabaticity effects [@lamagna]. Near the strong coupling limit the mobile $S1$ bipolaron has an effective mass of the order of a single Holstein polaron mass, so that one should not rule out the possibility of a superconducting state of $S1$ bipolarons with $s$ or $d$-wave symmetry in HHM [@trugman]. More recent diagrammatic Monte Carlo study [@Macridin] found $S1$ bipolarons for large $U$ at intermediate and large EPI and established the phase diagram of 2D HHM, comprising unbound polarons, $S0$ and $S1$ domains. Ref. [@Macridin] emphasised that the transition to the bound state and the properties of the bipolaron in HHM are very different from bound states in the attractive (negative $U$) Hubbard model without EPI [@micnas1981].

In any case the Holstein model is an extreme polaron model, with typically highest possible values of the (bi)polaron mass in the strong coupling regime [@ale5; @alekor; @spencer; @jim]. Many doped ionic lattices, including cuprates, are characterized 