  In the strangeness $S=-2$ sector, we study the $\Xi^{-}nn$ ($I=3/2,
J^P={1/2}^+$) three-body system using pionless halo effective field theory
(EFT), which provides a systematic model independent framework for assessing
the feasibility of light particle-stable three-body bound states, utilizing
low-energy universality. Here we take recourse to a simplistic speculation of
the three-body system by eliminating the repulsive spin-singlet $\Xi^- n$
sub-system, while retaining the predominantly attractive (possibly bound)
spin-triplet $\Xi^{-}n$ and the virtual bound spin-singlet $nn$ sub-system. In
particular, a qualitative leading order EFT investigation by introducing a
sharp momentum (ultraviolet) cut-off parameter $\Lambda_c$ into the
Faddeev-like coupled integral equations, indicates a discrete scaling behavior
akin to a renormalization group limit cycle, thereby suggesting the formal
existence of Efimov states in the unitary limit, as $\Lambda_c\to \infty$. Our
subsequent non-asymptotic analysis indicates that the three-body binding energy
$B_3$ is sensitively dependent on the cut-off without the inclusion of
three-body contact interactions. Furthermore, our analysis reproduces several
values of the binding energy $B_3\sim 3-4$ MeV, predicted in context of
existing potential models, with the regulator $\Lambda_c$ in the range, $\sim
350-460$ MeV. Finally, based on these model inputs for $B_3$, a ballpark
estimate of the three-body scattering length in the range, $2.6-4.9$ fm, is
naively constrained by our EFT analysis, ostensibly demonstrating the universal
nature of three-body correlations that is likely to manifest themselves in a
halo-bound $\Xi^{-}nn$ system.
