Effect of thresholds on the width of three-body resonances

It has been recently reported an intriguing theoretical result of a narrow three-body resonance with a large available phase space. The resonance was reported in the $N\Lambda\Lambda-\Xi NN$ system near the $\Xi d$ threshold, having a very small width in spite of the open $N\Lambda\Lambda$ channel lying around 23 MeV below the $\Xi NN$ channel. We use first-order perturbation theory as a plausible argument to explain this behavior. We apply our result to realistic local interactions. Other systems involving several thresholds are likely to follow the same behavior.