Schauder's Conjecture: "Every continuous function, from a nonempty compact and convex set in a (Hausdorff) topological vector space into itself, has a fixed point." [Problem 54 in The Scottish Book]
I wonder whether this conjecture is resolved. I know R. Cauty [Solution du problème de point fixe de Schauder, Fund. Math. 170 (2001) 231–246] proposed an answer, but apparently in the international conference "Fixed Point Theory and its Applications" in 2005, T. Dobrowolski remarked that there is a gap in the proof.