A counterexample to the spectral-uniqueness locus condition

Let $A=\operatorname{diag}(S,\tfrac12 J)$, where $S$ is the $2\times2$ swap matrix and $J$ the $2\times2$ all-ones matrix. It is a real symmetric, entrywise nonnegative, stochastic $4\times4$ matrix with spectrum $\{1,1,0,-1\}$ and trace one, and it is spectrally unique up to permutation. For a symmetric stochastic $B$ with this spectrum, the multiplicity of the eigenvalue one equals the number of connected components of the graph of $B$. A component carrying the eigenvalue $-1$ is bipartite with two classes of equal size. So $B$ has two components of order two, and these must be $S$ and $\tfrac12 J$. As a nontrivial convex combination of two symmetric stochastic matrices, $A$ is not a vertex of the polytope. Its zero entries keep it off every segment $[I,V]$, $[C_4,V]$ and $[I,C_4]$. Hence the conjectured necessary condition for spectrally unique matrices of positive trace fails for $n=4$.