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There was a long standing conjecture stating that the geometric location of eigenvalues of doubly stochastic matrices of order $n$ is exactly the union of regular $k$-gons anchored at $1$ in the unit disc for $2 \leq k \leq n$.

Mashreghi and Rivard showed that this conjecture is wrong for $n = 5$, cf. Linear and Multilinear Algebra, Volume 55, Number 5, September 2007 , pp. 491-498.

Have we made progress since then, beyond $n=5$, or for $n=4$? ($n=2,3$ is pretty simple).

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I don't know. Math Reviews shows only one paper citing Mashreghi & Rivard, and judging by the review that paper doesn't go in the direction you ask about. If after a while you get no answer here, I think your question would be appropriate for MathOverflow (being careful to notify both sites of the double-posting). – Gerry Myerson Nov 9 '11 at 3:11
Thanks Gerry. I'm not very familiar with what's more appropriate for MathOverflow vs. MathStackExchange, but I'll follow your advice. – Nathan Portland Nov 9 '11 at 3:29
Your question is perfectly appropriate here, and you may well get a superb answer here - I certainly hope so. I'm just saying if nothing materializes here in a couple of days.... – Gerry Myerson Nov 9 '11 at 3:48

I accidentally saw this paper today on the internet concerning the case $n=4$:

Jeremy Levick, Rajesh Pereira and David W. Kribs (2015), The four-dimensional Perfect-Mirsky Conjecture, Journal: Proc. Amer. Math. Soc., 143: 1951-1956.

Abstract: We verify the Perfect-Mirsky Conjecture on the structure of the set of eigenvalues for all $n\times n$ doubly stochastic matrices in the four-dimensional case. The $n=1,2,3$ cases have been established previously and the $n=5$ case has been shown to be false. Our proof is direct and uses basic tools from matrix theory and functional analysis. Based on this analysis we formulate new conjectures for the general case.

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