Minimax theorem for the spectral radius of the product of non-negative matrices

We prove the minimax equality for the spectral radius $\rho (AB)$ of the product of matrices $A\in \mathcal{A}$ and $B\in \mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are compact sets of non-negative matrices of dimensions $N\times M$ and $M\times N$, respectively, satisfying the so-called hourglass alternative.