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The K$^{-}$ over K$^{+}$ multiplicity ratio is measured in deep-inelasticscattering, for the first time for kaons carrying a large fraction $z$ of thevirtual-photon energy. The data were obtained by the COMPASS collaborationusing a 160 GeV muon beam and an isoscalar $^6$LiD target. The regime ofdeep-inelastic scattering is ensured by requiring $Q^2>1$ (GeV/$c)^2$ for thephoton virtuality and $W>5$ GeV/$c^2$ for the invariant mass of the producedhadronic system. Kaons are identified in the momentum range from 12 GeV/$c$ to40 GeV/$c$, thereby restricting the range in Bjorken-$x$ to $0.01<x<0.40$. The$z$-dependence of the multiplicity ratio is studied for $z>0.75$. For verylarge values of $z$, $i.e.$ $z>0.8$, we observe the kaon multiplicity ratio tofall below the lower limits expected from calculations based on leading andnext-to-leading order perturbative quantum chromodynamics. Also, the kaonmultiplicity ratio shows a strong dependence on the missing mass of thesingle-kaon production process. This suggests that within the perturbativequantum chromodynamics formalism an additional correction may be required,which takes into account the phase space available for hadronisation.

K$^{-}$ over K$^{+}$ multiplicity ratio for kaons produced in DIS with a large fraction of the virtual-photon energy