common feature of systems with active $\{xz, yz\}$ orbital degrees of freedom [@Harris2004]. As for the value of the hopping element $t$ in Ca$_2$RuO$_4$ we choose $t= 123$ meV [@Gorelov2010], i.e. $t = 22J$ for the above chosen realistic value of $J=5.6$ meV. We note that to simplify the analysis we skip here the spin-orbit coupling between holes in the $xz$ and $yz$ orbitals. Such a simplification is not [*a priori*]{} justified for a realistic situation in Ca$_2$RuO$_4$ but is rationalised by the intuitive understanding of its spectral functions presented in Ref. , which relies on the Hund’s coupling and does not include the spin-orbit coupling as an essential part. Moreover, a (surprisingly) good agreement between the theoretical results presented below and the experimental results, cf. Fig. \[result\], [*a posteriori*]{} legitimizes this assumption further. Finally, it will not affect the study of the possible intrinsic differences between the $S=1/2$ and $S=1$ spin polaron models.

There are two projections in place in the kinetic Hamiltonian (\[model2\]). First, due to the strong on-site Coulomb repulsion $U$–and since we confine ourselves to the low energy physics valid for energies smaller than the ‘Hubbard’ $U$–we restrict the hole motion to the Hilbert space spanned 