  For decades, it has remained an open question, whether two dopants in the
$t$--$J$ model bind in the strongly correlated regime of small spin couplings
versus hopping, $J \ll t$. Here, we investigate this problem in Bethe lattice
structures with Ising coupling $J_z$, and mainly focus on the case of a
coordination number equal to 4. The special geometry circumvents subtle effects
in regular lattices, but importantly still contains the non-trivial dependency
on the rotational symmetry and particle statistics. It, furthermore, allows us
to reach numerical convergence, and we conclusively answer whether binding
occurs or not. In particular, we find that the rotationally symmetric $s$ waves
unbind below $J_z \simeq 0.3 t$, which we unveil is tied to Pauli blocking of
symmetrical hole configurations. This is further substantiated by the fact that
holes in a bosonic spin environment shows strong -- superlinear -- binding at
low $J_z / t$. Additionally, higher rotational fermionic $p$ and $d$ waves
partially overcome the blocking, are perfectly degenerate, and bind for all
$J_z / t$. In the strongly correlated regime of $J_z \ll t$, this binding
occurs sublinearly with scaling $\sim t (J_z / t)^{3/2}$.
