with spin-polarized and particle imbalanced components consisting of $N_A=3$ and $N_B=1$ fermions. The mixture is confined within a double-well and it is initialized in its corresponding interspecies interacting ground state as described by the Hamiltonian of Eq. (\[Hamilt\]), using either imaginary time propagation or improved relaxation [@ML-MCTDHX] within ML-MCTDHX. The double-well is characterized by frequency $\omega=0.1$, barrier height $V_0=2$ and width $w=1$.

To inspect the ground state of the FF mixture we invoke the $\sigma$-species single-particle density $\rho_\sigma^{(1)}(x;t)$ \[see also Eq. (\[one\_body\_cor\])\]. Within the weakly interacting regime, $g_{AB}=0.1$, we observe that each $\rho_\sigma^{(1)}(x)$ shows an equal population in the two wells of the double-well and it is distributed in a symmetric manner both in the HF approximation as well as on the MB level, see Figs. \[abb:gs3-1\] (a) and (b). Also, $\rho_A^{(1)}(x;t)$ and $\rho_B^{(1)}(x;t)$ feature a miscible behavior in both approaches. Note that due to the particle imbalance the $A$-species which contains the higher particle number exhibits a broader single-particle density distribution within each well when compared to the $B$-species. Turning to the strong interaction regime, $g_{AB}=4.0$, an intrawell fragmentation of $\rho_\sigma^{(1)}(x)$ occurs on the MB level, see Fig. \[abb:gs3-1\] (a), while $\rho_A^{(1)}(x;t)$ and $\rho_B^{(1)}(x;t)$ show again a miscible behavior. 