The symmetric doubly stochastic matrix spectral determination conjecture

Let $\Delta_n^s$ denote the set of symmetric doubly stochastic $n\times n$ matrices, let $\Delta_n^s(a)$ denote the subset with the relevant parameter $a$, let $I_n$ be the identity matrix, let $C_n$ be the matrix defined in the paper, and let $P$ be a vertex of $\Delta_n^s$. For points $X,Y$, write $[X,Y]$ for the line segment joining them.

Spectral determination conjecture. For $0<a\leq n$, the only elements of $\Delta_n^s(a)$ that are DS in $\Delta_n^s$ are points on the line segments $[I_n,C_n]$, $[I_n,P]$ and $[C_n,P]$ where $P$ is a vertex of $\Delta_n^s$.

This conjecture proposes a complete description of the elements of $\Delta_n^s(a)$ that are determined by their spectra within the symmetric doubly stochastic matrices, extending the verified case $n=3$.