Bound the spectral radius of a positive matrix

Question

Let $A$ and $D_{0}$ be substochastic matrices (with non-negative elements and row sums less than 1), with $D_{0}$ being diagonal, $\iota$ be a vector of ones, and let $$M_{1}\equiv I-A,$$$$M_{2}\equiv I-D_{0}A^{T}.$$ Let $\mathcal{D}\left(v\right)$ be the diagonal matrix constructed from vector $v$. We want to show that $$\rho\left(\left(\mathcal{D}\left\{ M_{2}^{-1}\iota\right\} \right)^{-1}M_{2}^{-1}\left(I-D_{0}\right)D_{0}\mathcal{D}\left\{ A^{T}M_{2}^{-1}\iota\right\} M_{2}^{-T}\mathcal{D}\left(M_{1}\iota\right)\right)<1.$$ Does somebody have an idea to prove it or of whether it holds at all? If it might not generally hold, what could a potential counterexample feature? Thanks in advance!