  The Bayesian reweighting procedure is extended to the case of multiple
independent extractions of transverse momentum dependent parton distributions
(TMDs). By exploiting the data on transverse single spin asymmetries, $A_N$,
for inclusive pion production in polarized proton-proton collisions measured at
RHIC, we perform a simultaneous reweighting of the quark Sivers, transversity
and Collins TMD functions extracted from semi-inclusive deep inelastic
scattering (SIDIS) and $e^+ e^-$ annihilation into hadron pairs. The impact of
the implementation of the Soffer bound, as well as the differences between
older and newer $A_N$ data, are investigated. The agreement with $A_N$ data at
large-$x_F$ values, a kinematical region complementary to those explored in
SIDIS measurements, is enhanced, improving the knowledge of the polarized quark
TMDs in the large-$x$ region.
