Strong anti-diagonal eigenvalue property conjecture for stochastic matrices

Let $A$ be a stochastic matrix with distinct eigenvalues. Say that $A$ has the strong anti-diagonal eigenvalue property when it satisfies the stronger anti-diagonal eigenvalue condition defined in the paper, and let $H^\lambda$ denote the parameterized family of lower-triangular matrices introduced there. Strong anti-diagonal eigenvalue property conjecture. $A$ has the strong anti-diagonal eigenvalue property if and only if it is of the form $H^\lambda$. The conjecture would classify stochastic matrices with distinct eigenvalues satisfying this strengthened spectral property; the paper states that it can be proved for $n\leqslant4$, while the general case is left open.