which can be understood from Eq. (\[exTBHM\_MI2\]). As seen from the figure, the finite $U_{12}$ enhances the DW(1,0) lobe. The finite $U_{12}$ also lifts the degeneracy of the DW states, and the state with $n^1_{p,q}=n^2_{p,q}$ has higher energy. So, the density of the DW states with finite $U_{12}$ has $n^{1,A}=n^{2,B}$ and $n^{1,B}=n^{2,A}$.

The MI(1,1) lobe remains unchanged in size, but, it is shifted upward in the phase diagram. The shift is attributed to the increase in effective chemical potential arising from the interaction energy associated with finite $U_{12}$. Similar trend, enhancement of DW(1,0) lobe, occur in the case of $U_{12}=0$ on increasing $V_k$. In addition to the MI phase, the DW(2,1) and similar DW phase with non-zero $n^{k,A}$ and $n^{k,B}$ are also energetically disfavoured. However, the most important feature is the emergence of prominent SS phase envelope around each of the DW lobes. On increasing $U_{12}$ further, as seen from the Fig. \[ebhm\_v12\_0\] (c-d), the MI lobes are transformed into DW lobes. And, at higher $U_{12}$ only the DW(n,0) phase, with $n\in\mathbb{N}$, are present in the system. The domain of the SS phase also increases. Ultimately, the SS envelopes around each of DW lobes merge into single large SS domain 