We investigate asphericity of the relative group presentation $$\langle G,t\mid atbtctdtet=1\rangle$$ and prove it aspherical provided the subgroup of $G$ generated by $$\{ ab^{-1}, bc^{-1}, cd^{-1}, de^{-1}\}$$ is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation $$\langle G,s,t \mid \alpha s\beta s\gamma t= 1=\delta t\varepsilon t\zeta s^{-1}\rangle. 2000 Mathematics Subject Classification: 20F05, 57M05.