It's not really about proving a negative. It has much more to do with how thorough search efforts have been and whether there have been any credible sightings.
If you compare and contrast the IUCN entries for the C. Principalis and C. Imperialis, you'll find that they're vastly different. Here are two of the most salient passages for the IBWO:
"Strong claims for this species's persistence in Arkansas and Florida have emerged since 2004 although the evidence remains highly controversial."
"Between the last confirmed sightings in 1944 and the 2004 records discussed above there were a further 20 credible unconfirmed reports from within its historic range. The species is considered likely to be extinct in Cuba, as intensive searches have not found any new records subsequent to those of the late 1980s." (Citations omitted.)
With regard to the Imperial:
"This species has not been recorded with certainty since 1956, and extensive habitat destruction and fragmentation combined with hunting may well have driven the species extinct. Extensive and prolonged searches within its former range (often following up on anecdotal reports) have failed to confirm the persistence of any individuals. Thorough mapping and analysis of remaining habitat has been conducted and the results do not provide much hope that any population has been able to survive. However, it cannot yet be presumed to be Extinct as the degree to which individuals can utilise sub-optimal regenerating forest is unknown, and it remains possible that some individuals survive. Any remaining population is likely to be tiny, and for these reasons it is treated as Critically Endangered (Possibly Extinct). "
