TITLE: Is Schauder's Conjecture Resolved?
QUESTION [43 upvotes]: 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.

REPLY [4 votes]: There is  R. Catty paper from 2012 titled 'Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes' which from what I heard was reviewed to establish a  correct proof of Schauder's conjecture ( or rather a generalization for iterates of f ), and will appear in an international journal.