The doubly stochastic versus permutation binary X-ray conjecture

Let $D$ be a doubly stochastic matrix of order $n$, meaning that it has nonnegative real entries and every row and column sums to $1$. Its binary diagonal X-ray is the vector $d(D)=(d_1,d_2,\dots,d_{2n-1})$ whose entries record which diagonals contain nonzero mass; for a permutation $\pi \in S_n$, write $d(\pi)$ for the analogous X-ray. The doubly stochastic versus permutation binary X-ray conjecture. There is a doubly stochastic matrix $D$ of order $n$ with binary diagonal X-ray

$$d(D)=(d_1,d_2,\dots,d_{2n-1})$$

if and only if there is a permutation $\pi \in S_n$ with binary diagonal X-ray

$$d(\pi)=(d_1,d_2,\dots,d_{2n-1}).$$

The forward implication asserts that every binary diagonal pattern feasible for a doubly stochastic matrix is already feasible for a permutation matrix. The density conditions are necessary by the Birkhoff decomposition, while sufficiency remains conjectural.