Symmetry and Hankel structure of minimal matrices of fixed RSK shape

Let $\lambda$ be a partition with $n$ parts, and let $M=(m_{i,j})_{1\le i,j \le n}\in\mathcal{M}_\lambda$ be a minimal matrix of shape $\lambda$. A matrix is Hankel when its entries depend only on the sum of their indices. Symmetry and Hankel conjecture. Every minimal matrix $M$ of shape $\lambda$ is symmetric,

$$m_{i,j}=m_{j,i}\quad\text{for all }1\le i,j\le n,$$

and is Hankel: there exists a sequence of integers $s_2,s_3,\dots,s_{2n}$ such that

$$m_{i,j}=s_{i+j}\quad\text{for all }1\le i,j\le n.$$

The conjecture extends the characterization of minimum-inversion permutations of fixed RSK shape to nonnegative integer matrices of the smallest possible size. It is proved in the paper for two-row partitions, while the general case remains open.