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We demonstrate that triangle singularity (TS) and box singularity (BS)mechanisms can produce unique narrow enhancements at the $\Lambda_c\bar{D}$ and$\Lambda_c\bar{D}^*$ thresholds in the invariant mass spectra of $J/\psi p$ and$J/\psi p\pi$, respectively. Taking into account that such mechanisms onlydepend on the initial $\Sigma_c^{(*)}\bar{D}^{(*)}$ interactions near thresholdwithin the TS or BS kinematic regimes, the $\Lambda_c\bar{D}$ and$\Lambda_c\bar{D}^*$ threshold enhancements can be regarded as a feed-downphenomenon originated from both the heavier pentaquark decays and the$\Sigma_c^{(*)}\bar{D}^{(*)}$ scatterings from the continuum. A search forthese structures in the $J/\psi p$ and $J/\psi p\pi$ spectra in both exclusiveand semi-inclusive processes will provide a smoking-gun evidence for thehadronic molecule nature of those observed pentaquarks and clarify the roleplayed by the TS and BS in the near-threshold dynamics.

Predictions for feed-down enhancements at the $\Lambda_c \bar{D}$ and $\Lambda_c \bar{D}^*$ thresholds via the triangle and box singularities