  We present a phenomenological study on the role of charm contribution and
$SU(2)$ isospin symmetry in the extraction of the $\Lambda$ polarizing
fragmentation functions from $e^+e^- \to \Lambda^\uparrow
(\bar\Lambda^\uparrow) \,h + X$ annihilation processes. We adopt the
well-established transverse-momentum-dependent factorization formalism, within
the Collins-Soper-Sterman evolution scheme at next-to-leading logarithm
accuracy, carefully exploiting the role of the nonperturbative component of the
polarizing fragmentation function. We then discuss the impact of these results
on the predictions for transverse $\Lambda$, $\bar{\Lambda}$ polarization in
semi-inclusive deep inelastic scattering processes at typical energies of the
future Electron-Ion Collider.
