  The reaction $K^-d \to \pi \Sigma n$ is studied within a Faddeev-type
approach, with emphasis on the specific kinematics of the E31 experiment at
J-PARC, i.e. $K^-$ beam momentum of $p_K = 1$ GeV/c and neutron angle of
$\theta_n=0^\circ$. The employed Faddeev approach requires as main input
amplitudes for the two-body subsystems $\bar KN \to \bar KN$ and $\bar KN \to
\pi\Sigma$. For the latter results from recently published chiral unitary
models of the $\bar KN$ interaction are utilized. The $\bar KN \to \bar KN$
amplitude itself, however, is taken from a recent partial-wave analysis. Due to
the large incoming momentum of the $K^-$, the $\bar KN$ interaction is probed
in a kinematical regime where those chiral potentials are no longer applicable.
A comparison of the predicted spectrum for various $\pi \Sigma$ charge channels
with preliminary data is made and reveals a remarkable agreement as far as the
magnitude and the line shape in general is concerned. Noticeable differences
observed in the $\pi\Sigma$ spectrum around the $\bar KN$ threshold, i.e. in
the region of the $\Lambda$(1405) resonance, indicate a sensitivity to the
details of the employed $\bar KN \to \pi\Sigma$ amplitudes and suggest that
pertinent high-precision data could indeed provide substantial constraints on
the structure of the $\Lambda$(1405).
