  We derive a connection between the fourth coefficient of the short-distance
Taylor expansion of the one-body correlation function, and the local three-body
correlation function of the Lieb-Liniger model of $\delta$-interacting spinless
bosons in one dimension. This connection, valid at arbitrary interaction
strength, involves the fourth moment of the density of quasi-momenta.
Generalizing recent conjectures, we propose approximate analytical expressions
for the fourth coefficient covering the whole range of repulsive interactions,
validated by comparison with accurate numerics. In particular, we find that the
fourth coefficient changes sign at interaction strength $\gamma_c\simeq 3.816$,
while the first three coefficients of the Taylor expansion of the one-body
correlation function retain the same sign throughout the whole range of
interaction strengths.
