  In this note, we show that the classical Wallach manifold
$\mathrm{SU}(3)/\mathrm{T}^2$-admits metrics of positive intermediate Ricci
curvature $(\mathrm{Ric}_d >0)$ for $d = 1, 2, 3, 4, 5$ that lose these
properties under the homogeneous Ricci flow for $d=1, 2, 3, 5$. We make the
same analyses to the family of Riemannian flag manifolds
$\mathrm{SU}(m+2p)/\mathrm{S}(\mathrm{U}(m)\times\mathrm{U}(p)\times
\mathrm{U}(p))$, concluding similar results. These explicitly verify some
claims expected to be true among experts (see \cite{BW}) for positive Ricci
curvature and intermediate positive Ricci curvature. Our technique is only
possible due to the global behavior understanding of the homogeneous Ricci flow
for invariant metrics on these manifolds.
