Eigenvalues of doubly stochastic matrices

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).

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.