Matrices with different Gondran–Minoux and determinantal ranks over $\mathrm{max}$-algebras

Let $\mathrm{GMr}(A)$ be the row Gondran–Minoux rank of a matrix, $\mathrm{GMc}(A)$ be the column Gondran–Minoux rank, and $\mathrm d(A)$ be the determinantal rank, respectively. The following problem was posed by M. Akian, S. Gaubert, and A. Guterman: Find the minimal numbers $m$ and $n$ such that there exists an $(m\times n)$-matrix $B$ with different row and column Gondran–Minoux ranks. We prove that in the case $\mathrm{GMr}(B)>\mathrm{GMc}(B)$ the minimal $m$ and $n$ are equal to 5 and 6, respectively, and in the case $\mathrm{GMc}(B)>\mathrm{GMr}(B)$ the numbers $m=6$ and $n=5$ are minimal. An example of a matrix $A\in\mathcal M_{5\times6}(\mathbb R_\mathrm{max})$ such that $\mathrm{GMr}(A)=\mathrm{GMc}(A^\mathrm t)=5$ and $\mathrm{GMc}(A)=\mathrm{GMr}(A^\mathrm t)=4$ is provided. It is proved that $p=5$ and $q=6$ are the minimal numbers such that there exists an $(p\times q)$-matrix with different row Gondran–Minoux and determinantal ranks.