  The family of invariant Riemannian manifolds in the Wallach flag manifold
$\mathrm{SU}(3)/\mathrm{T}^2$ is described by three parameters $(x,y,z)$ of
positive real numbers. By restricting such a family of metrics in the
\emph{tetrahedron} $\cal{T}:= x+y+z = 1$, in this paper, we describe all
regions $\cal R \subset \cal T$ admitting metrics with curvature properties
varying from positive sectional curvature to positive scalar curvature,
including positive intermediate curvature notion's. We study the dynamics of
such regions under the \emph{projected Ricci flow} in the plane $(x,y)$,
concluding sign curvature maintenance and escaping. In addition, we obtain some
results for positive intermediate Ricci curvature for a path of metrics on
fiber bundles over $\mathrm{SU}(3)/\mathrm{T}^2$, further studying its
evolution under the Ricci flow on the base.
