  This work provides the first combined analysis of low-energy p$\Lambda$
scattering, considering both cross section and correlation data. The obtained
results establish the most stringent constraints to date on the two-body
p$\Lambda$ interaction, pointing to a weaker attraction than so far accepted.
The best set of scattering lengths for the spin singlet and triplet are found
to range from $f_0, f_1 = (2.1, 1.56)$ to $(3.34, 1.18)~$fm. With a chiral NY
potential fine-tuned to those scattering parameters, the in-medium properties
of the $\Lambda$ are explored and a potential depth of $U_\Lambda= -36.3\pm 1.3
\mathrm{(stat)}^{+2.5}_{-6.2}\mathrm{(syst)}$ MeV is found at nuclear matter
saturation density.
