A Proof of the Dittert Conjecture via Boundary Rigidity

Let $\mathcal{K}_n$ be the simplex of nonnegative $n \times n$ matrices whose entries sum to $n$. For $A$ in $\mathcal{K}_n$, with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$, define $Φ(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\mathrm{per}(A)$. Dittert conjectured that $Φ$ is uniquely m