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).