Status of the Hadamard Circulant conjecture The following feels like a community wiki question, so I do it here:
Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser
(a long standing difficult conjecture):
There is no Circulant Hadamard matrix with more than $4$ columns
see
Barry Hurley, Paul Hurley, Ted Hurley, The Hadamard circulant conjecture, Bulletin of the London Mathematical Society, Volume 44, Issue 1, February 2012, Page 206, https://doi.org/10.1112/blms/bdq112
More recently  a gap was found in the proof:
see
https://www.sfu.ca/~jed/Papers/Craigen%20Jedwab.%20Circulant%20Hadamard.%20Preprint.pdf
Question: What is the real status of this Conjecture ?
 A: The paper by Craigen and Jedwab points out a very definite flaw in the main theorem by Hurley, Hurley & Hurley, providing a counterexample to that theorem.  So the conjecture is still open.
The paper by Leung and Schmidt (here) gives the latest information about possible counterexamples to the Circulant Hadamard conjecture.
A website of Bernhard Schmidt lists smallest open cases based (I believe) on the work of the paper cited above.  The smallest open cases are, of course, quite large.
A: Unfortunatly, there has been confusion about the status of this problem for decades. 
All correct results on the circulant Hadamard matrix conjecture are contained in the following papers:
[1] Turyn, Richard J.: Character sums and difference sets. Pacific J. Math. 15 1965 319–346.
[2] Leung, Ka Hin; Schmidt, Bernhard: The field descent method. Des. Codes Cryptogr. 36 (2005), no. 2, 171–188.
[3] Leung, Ka Hin; Schmidt, Bernhard: New restrictions on possible orders of circulant Hadamard matrices. Des. Codes Cryptogr. 64 (2012), no. 1-2, 143–151.
Computational results and the latest status on the smallest open cases can be found in
[4] Peter Borwein, Michael J. Mossinghoff: Wieferich pairs and Barker sequences, II. 
arXiv:1306.0045v2
Bernhard Schmidt
A: Craigen and Jedwab have indicated that the September 2011 revision of Hurley, Hurley & Hurley also has a flaw:
https://www.sfu.ca/~jed/Papers/Craigen%20Jedwab.%20Circulant%20Hadamard%202.%20Preprint.pdf
A: About two weeks ago, at the International Workshop on Hadamard Matrices and Their Applications (RMIT), http://user.gs.rmit.edu.au/asha/iwhma/ Jennifer Seberry presented a proof, based on counting arguments http://user.gs.rmit.edu.au/asha/iwhma/html/Wednesday/JenniferSeberry.pdf (Wayback Machine), but that proof was quickly found to be incomplete. As far as I know, she is currently working on improving the proof. A special volume of the Australasian Journal of Combinatorics http://user.gs.rmit.edu.au/asha/iwhma/html/ajc.html is forthcoming, and with any luck, things will become much clearer at that point.
A: On sep 4, 2011, authors Hurley, Hurley & Hurley of the paper on the Circulant Hadamard conjecture, has posted the same paper on the arxiv.  Did they corrected the error reported by Craigen and Jedwab on feb 10, 2011?
A: check:
Barry Hurley, Paul Hurley, Ted Hurley, The Hadamard circulant conjecture, Bulletin of the London Mathematical Society, Volume 44, Issue 1, February 2012, Page 206, https://doi.org/10.1112/blms/bdq112
A: I recently found out about this conjecture and found this paper from 2019 on arXiv,
https://arxiv.org/pdf/1907.12683.pdf
As far as I can tell, this paper is still unpublished and on the Wikipedia-page, it says that Circulant Matrix Conjecture is still an open problem. Is the proof in the aforementioned paper known to have gaps?
