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We study for the first time the pΣ− → K−d and its time reversal reaction close to threshold and show that they are driven by a triangle mechanism, with the Λ(1405), a proton and a neutron as intermediate states, which develops a triangle singularity close to the K̅d threshold. As the main result, we find that the cross section, well within measurable range, is very sensitive to different models that, while reproducing K̅N observables above threshold, provide different extrapolations of the K̅N amplitudes below threshold. The observables of this reaction will provide new constraints on the theoretical models, leading to more reliable extrapolations of the K̅N amplitudes below threshold and to more accurate predictions of the Λ(1405) state of lower mass.