Not unresolved today, but...

The Lusternik-Schnirelmann Theorem of the three geodesics claims that on any Riemannian manifold topologically a sphere there are three simple (non-self-intersecting) closed geodesics (and for the ellipsoid, only three). This was "proved" in 1929, but the proof was soon recognized to be flawed, and was not entirely settled until ~50 years later, in 1978, followed by another proof and generalization in 1985:

Werner Ballmann, "Der Satz von Lusternik und Schnirelmann" Math. Shriften 102, 1-25, 1978.

Wilhelm Klingenberg, "The existence of three short closed geodesics", Differential geometry and complex analysis, Springer, Berlin, pp. 169–179, 1985.

Not unresolved today, but...

The Lusternik-Schnirelmann "Theorem of the three geodesics" claims that on any Riemannian manifold topologically a sphere there are three simple (non-self-intersecting) closed geodesics (and for the ellipsoid, exactly three). This was "proved" in 1929, but the proof was soon recognized to be flawed. It was not entirely settled until ~50 years later, by Ballmann in 1978, followed by another proof and generalization by Klingenberg in 1985:

Werner Ballmann, "Der Satz von Lusternik und Schnirelmann" Math. Shriften 102, pp. 1-25, 1978.

Wilhelm Klingenberg, "The existence of three short closed geodesics," Differential Geometry and Complex Analysis, Springer, Berlin, pp. 169–179, 1985.