On question about semi-modularity of lattice of subgroups of finite groups

This article considers finite groups whose lattice of subgroups satisfy certain generalized semi-modularity conditions. The main result is the theorem: the lattice of subgroups of the finite group $G$ is $1$-lower semi-modular whenever the lattice of subgroups of $G$ is upper semi-modular and the lattice of subgroups of any proper subgroup of $G$ is lower semi-modular.