Generalized eigenvectors of block triangular random matrices

Let $$A = (a_{ij})_{1 \leq i,j \leq N}$$ and $$B = (b_{ij})_{1 \leq i,j \leq N}$$ be random matrices, with each $$a_{ij}$$ and $$b_{ij}$$ an independent random variable with continuous density function, zero mean and bounded variance. Let $$\textbf{c}=(c_j)_{1 \leq j \leq N}$$ be a random vector, whose entries are also independent and have a continuous, zero mean, bounded variance distribution.

Define the matrix $$D$$ by $$D = \begin{pmatrix} A & 0 \\ BA &A\end{pmatrix}$$ and let $$\textbf{e} = \begin{pmatrix} \textbf{c} \\ B\textbf{c}\end{pmatrix}$$

Let $$\textbf{q}_i$$ be the generalized eigenvectors of $$D$$ for $$1 \leq i \leq 2\cdot N$$. What can we say about $$\textbf{q}_i \cdot \textbf{e}$$? In particular, is it possible to prove (or disprove) that $$P \left( \bigcup_i \textbf{q}_i \cdot \textbf{e} \neq 0\right)=1$$? That is, that $$\textbf{e}$$ is not orthogonal to any of the generalized eigenvectors of $$D$$ with probability one?