I just noticed this sentence:
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With support from the Arkansas Audubon Society Trust, he [Luneau] organized and led a less-extended expedition in January of 2003 to look for ivory-bills in Arkansas’ White River National Wildlife Refuge.
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That means that "prime-time" Arkansas expeditions in early 2003, 2004, and 2005 have already been completed with no hard evidence of an Ivory-bill. I think it's unlikely that the core problem here is "not enough searching", and I think it's unlikely that field season #4 will finally yield the definitive photograph(s).
Remember, some 20,000 hours have already been expended since Sparling's initial report, and the reported sightings occurred in a small (four-square-kilometer) area.
In that small area, if you were ever going to gather conclusive proof of an Ivory-bill, I think you'd probably already have it after the first couple hundred hours of "prime time" searching. By the time you'd racked up 1,000 or 2,000 hours without hard evidence, I think it would be sensible to question whether the bird was present. If you expanded the search area and racked up five, ten, or twenty thousand hours without hard evidence, at some point you'd need to face the real prospect that your glimpses, fuzzy pictures, and audio may all be "false positives".