  We compute the gluon Sivers function $f_{1T}^{\perp g}(x,k_\perp)$ of the
transversely polarized nucleon at small-$x$ by exploiting the known connection
between the dipole gluon Sivers function and the Odderon. We numerically solve
the evolution equation for the Odderon both in the linear and nonlinear
regimes. While we find that the $x$ and $k_\perp$ dependences of the Sivers
function do not factorize as a result of the quantum evolution, factorization
breaking is not numerically significant, and is much milder than what one
expects in the case of unpolarized TMDs. We also point out the possibility
that, due to the presence of a node in the Sivers function, single spin
asymmetry for open charm production in semi-inclusive deep inelastic scattering
flips signs as the transverse momentum of D-mesons is varied. This can be
tested at the future Electron-Ion Collider.
