  We propose a conjectural list of Fano manifolds of Picard number $1$ with
pseudoeffective normalized tangent bundles, which we prove in various
situations by relating it to the complete divisibility conjecture of Russo and
Zak on varieties with small codegree. Furthermore, the pseudoeffective
thresholds and hence the pseudoeffective cones of the projectivized tangent
bundles of rational homogeneous spaces of Picard number $1$ are explicitly
determined by studying the total dual VMRT and the geometry of stratified Mukai
flops. As a by-product, we obtain sharp vanishing theorems on the global
twisted symmetric holomorphic vector fields on rational homogeneous spaces of
Picard number $1$.
