  Quarkonium production mechanism in high multiplicity small collision systems
has recently been pursued in the color-glass-condensate (CGC) effective theory
combined with non-relativistic QCD (NRQCD) factorization, allowing to study
initial state interactions. Quarkonium polarization, potentially measured in
future experiments, would help elucidate the quarkonium production mechanism at
high multiplicities. In this paper, we provide predictions on $J/\psi$
polarization parameters in high multiplicity proton-proton ($pp$) and
proton-nucleus ($pA$) collisions within the CGC+NRQCD framework. Theoretical
predictions are given for $J/\psi$ rapidity $2.5< y_{J/\psi}<4$,
charged-particle multiplicity pseudorapidity $|\eta_{ch} | <1$ and energies
$\sqrt{S}=13\mathrm{~TeV}$ for $pp$, $\sqrt{S}=8.16\mathrm{~TeV}$ for $pA$
collisions. Considering two leptonic frame choices (Collins - Soper and
helicity) we find a weak polarization of $J/\psi$ that additionally decreases
with growing event activities. No significant system size dependence between
$pp$ and $pA$ collisions is obtained - this could be a new constraint to
initial state interactions in small collision systems.
