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We used the Atacama Pathfinder Experiment (APEX) telescope to search for CI 1-0 (492.16GHz) emission towards 8 proplyds in NGC 1977, which is an FUV radiation environment two orders of magnitude weaker than that irradiating the Orion Nebular Cluster (ONC) proplyds. CI is expected to enable us to probe the wind launching region of externally photoevaporating discs. Of the 8 targets observed, no 3$σ$ detections of the CI line were made despite reaching sensitivities deeper than the anticipated requirement for detection from prior APEX CI observations of nearby discs and models of external photoevaporation of quite massive discs. By comparing both the proplyd mass loss rates and CI flux constraints with a large grid of external photoevaporation simulations, we determine that the non-detections are in fact fully consistent with the models if the proplyd discs are very low mass. Deeper observations in CI and probes of the disc mass with other tracers (e.g. in the continuum and CO) can test this. If such a test finds higher masses, this would imply carbon depletion in the outer disc, as has been proposed for other discs with surprisingly low CI fluxes, though more massive discs would also be incompatible with models that can explain the observed mass loss rates and CI non-detections. The expected remaining lifetimes of the proplyds are estimated to be similar to those of proplyds in the ONC at 0.1Myr. Rapid destruction of discs is therefore also a feature of common, intermediate UV environments.