  The study of the strength and behaviour of the antikaon-nucleon
($\mathrm{\overline{K}N}$) interaction constitutes one of the key focuses of
the strangeness sector in low-energy Quantum Chromodynamics (QCD). In this
letter a unique high-precision measurement of the strong interaction between
kaons and protons, close and above the kinematic threshold, is presented. The
femtoscopic measurements of the correlation function at low pair-frame relative
momentum of (K$^+$ p $\oplus$ K$^-$ $\overline{\mathrm{p}}$) and (K$^-$ p
$\oplus$ K$^+$ $\overline{\mathrm{p}}$) pairs measured in pp collisions at
$\sqrt{s}$ = 5, 7 and 13 TeV are reported. A structure observed around a
relative momentum of 58 MeV/$c$ in the measured correlation function of (K$^-$
p $\oplus$ K$^+$ $\overline{\mathrm{p}}$) with a significance of 4.4. $\sigma$
constitutes the first experimental evidence for the opening of the
$(\mathrm{\overline{K}^0 n} \oplus \mathrm{K^0 \overline{n}})$ isospin breaking
channel due to the mass difference between charged and neutral kaons. The
measured correlation functions have been compared to J\"{u}lich and Kyoto
models in addition to the Coulomb potential. The high-precision data at low
relative momenta presented in this work prove femtoscopy to be a powerful
complementary tool to scattering experiments and provide new constraints above
the $\mathrm{\overline{K}N}$ threshold for low-energy QCD chiral models.
