  Non-Hermitian quantum one-parametric $N$ by $N$ matrix Hamiltonians
$H^{(N)}(\lambda)$ with real spectra are considered. Their special choice
$H^{(N)}(\lambda)=J^{(N)}+\lambda\,V^{(N)}(\lambda)$ is studied at small
$\lambda$, with a general $N^2-$parametric real-matrix perturbation
$\lambda\,V^{(N)}(\lambda)$, and with the exceptional-point-related
"unperturbed" Jordan-block Hamiltonian $J^{(N)}$. A "stability corridor" ${\cal
S}$ of the parameters $\lambda$ is then sought guaranteeing the reality of
spectrum and realizing a unitary-system-evolution access to the
exceptional-point boundary of stability. The corridors are then shown
$N-$dependent and "narrow", corresponding to certain specific,
unitarity-compatible perturbations with "admissible" matrix elements
$V^{(N)}_{j+k,j}(\lambda) ={\cal O}(\lambda^{(k-1)/2})\,$ at subscripts
$k=1,2,\ldots,N-1\,$ and at all $j$.
