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In this note, we show that the classical Wallach manifold$\mathrm{SU}(3)/\mathrm{T}^2$-admits metrics of positive intermediate Riccicurvature $(\mathrm{Ric}_d >0)$ for $d = 1, 2, 3, 4, 5$ that lose theseproperties under the homogeneous Ricci flow for $d=1, 2, 3, 5$. We make thesame 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 someclaims expected to be true among experts (see \cite{BW}) for positive Riccicurvature and intermediate positive Ricci curvature. Our technique is onlypossible due to the global behavior understanding of the homogeneous Ricci flowfor invariant metrics on these manifolds.

On the dynamics of positively curved metrics on $\mathrm{SU}(3)/\mathrm{T}^2$ under the homogeneous Ricci flow