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My impression is that Stanley's partionability conjecture and his depth conjecture (which was shown to imply the partionability conjecture in 2008) were both believed to be true, until Duval, Goeckner, Klivans, and Martin found a counterexample to the partionability conjecture in 2015. See this AMS survey article: https://www.ams.org/journals/notices/201702/rnoti-p117.pdf. This fits with a few other examples already mentioned (the Hauptvermutung, the Hirsch conjecture, ...) which warn us that although simplicial complexes/polytopes may appear to be intuitively simple objects, they can in fact be extremely complicated.