  We investigate the Beyond Standard Model discovery potential in the framework
of the Effective Field Theory (EFT) for the same-sign $WW$ scattering process
in purely leptonic $W$ decay modes at the High-Luminosity and High-Energy
phases of the Large Hadron Collider (LHC). The goal of this paper is to examine
the applicability of the EFT approach, with one dimension-8 operator varied at
a time, to describe a hypothetical new physics signal in the $WWWW$ quartic
coupling. In the considered process there is no experimental handle on the $WW$
invariant mass, and it has previously been shown that the discovery potential
at 14 TeV is rather slim. In this paper we report the results calculated for a
27 TeV machine and compare them with the discovery potential obtained at 14
TeV. We find that while the respective discovery regions shift to lower values
of the Wilson coefficients, the overall discovery potential of this procedure
does not get significantly larger with a higher beam energy.
