  The increase of strange-particle yields relative to pions versus
charged-particle multiplicity in proton-proton (pp) collisions at the LHC is
usually described by microscopic or hydrodynamical models as a result of the
increasing density of produced partons or strings and their interactions.
Instead, we consider the multiple partonic interaction (MPI) picture originally
developed in the context of the PYTHIA event generator. We find that
strangeness enhancement in PYTHIA is hidden by a large excess of low-$p_{\rm
T}$ multi-strange baryons, which mainly results from the hadronization of
$u$-quark, $d$-quark and gluon ($udg$) strings. Strange baryons produced in
strings formed from parton showers initiated by strange quarks
($s$-fragmentation), however, describe well the spectral shapes of $\Xi$ and
$\Omega$ baryons and their multiplicity dependence. Since the total particle
yield contains contributions from soft and hard particle production, which
cannot be experimentally separated, we argue that the correct description of
the $p_{\rm T}$-spectra is a minimum requirement for meaningful comparisons of
multiplicity dependent yield measurements to MPI based calculations. We
demonstrate that the $s$-fragmentation component describes the increase of
average $p_{\rm T}$ and yields with multiplicity seen in the data, including
the approximate multiplicity scaling for different collision energies. When
restricted to processes that reproduce the measured $p_{\rm T}$-spectra, the
MPI framework exhibits a smooth evolution from strictly proportional
multiplicity scaling ($K_{\rm S}^0$, $\Lambda$, where the $udg$-hadronization
component dominates) to linearity ($s$-fragmentation) and on to increasingly
non-linear behavior ($c$-, $b$-quark and high-$p_{\rm T}$ jet fragmentation),
hence providing a unified approach for particle production in pp collisions.
