In a recent study of extended TBHM (eTBHM) [@wilson_16], it was shown that the SS phase exists for small value of NN interaction. In this work, the NN interaction was limited to either one of the species or between the species. We address this research gap by including all the possible intra- and inter-species NN interactions. Such a model is apt to describe the physics of dipolar Bose-Bose mixtures in optical lattices. An example of such a combination is the recently realized Er-Dy mixture [@trautmann_18]. An important result of our work is the possibility to realize compressible and incompressible quantum phases with spatial segregation. Such a phase could be instrumental in examining superfluid instabilities and other non-equilibrium properties in the lattice models of quantum liquids.

The remaining of the paper is organized into four sections. In the Section \[sec\_theory\] we describe the zero temperature Hamiltonian of the TBHM and discuss the Gutzwiller mean-field theory of the model. We then discuss the mean-field decoupling theory to calculate the compressible-incompressible phase boundaries analytically. This is followed by a brief discussion on the characterization of quantum phases. The phase diagrams of TBHM are discussed in the Section \[sec\_phdiag\]. The Section \[sec\_phdiag\_ext\] discussed the 