Two hypotheses on the class $\mathcal{L}(γ)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are discussed. Two weak hypotheses on the class $\mathcal{L}(γ)$ in the class $\mathcal{OS}\cap \mathcal{ID}$ are proved. A necessary and sufficient condition in order that, for every $t >0$, the $t$-th convolution power of a distribution in the class $\mathcal{OS}\cap \mathcal{ID}$ belongs to the class $\mathcal{L}(γ)$ is given.