Is Schauder's conjecture resolved? 
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.
 A: There is  R. Cauty 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.
Edit: published: R. Cauty, J. Reine Angew. Math. 729 (2017), 1–27 DOI link
A: An old proof (by M. R. Taskovic) is here: Mathematica Moravica, volume 2, (1998).
Summary: " The most famous of many problems in nonlinear analysis is Schauder’s problem (Scottish book, problem 54) of the following form, that if $C$ is a nonempty convex compact subset of a linear topological space does every continuous mapping $f : C → C$ has a fixed point? The answer we give in this paper is yes."
A new proof (by Mohamed Ennassik & Mohamed Aziz Taoudi
) is  here: Journal of Fixed Point Theory and Applications volume 23, 52 (2021).
Summary: "In this paper, we prove that every nonempty compact $s$-convex subset ($0<s\le 1$) of a Hausdorff topological vector space has the fixed point property. Our approach allows us to provide a simple alternative proof of Schauder's conjecture."
A: I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:
Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.
In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.
A: In Points fixes des applications compactes dans les espaces ULC published in in the arXiv in 2010 Robert Cauty wrote
il y a d’ailleurs une erreur dans la demonstration du
lemme 3 de [2], qu’il n’y a plus de raison de corriger, vu la superiorite de la
nouvelle approche
(there is, by the way, an error in the proof of lemma 3 from [2] for which there is no need for correction in view of the superiority of the new approach)
It seems thus that Cauty still (or again) claims that the Schauder conjecture is settled.
[2] is: R. Cauty. Solution du problème de point fixe de Schauder. Fund. Math. 170, 2001, 231-246.

Edit (August 2016). I was quite surprised that apparently there is no version of the mentioned 2010 arXiv article published in an international journal. While searching the web I learned that Robert Cauty died in 2013.
