On $M$-divisibility of stable laws of stable laws

A one dimensional distribution $F$ is called $M$-infinitely divisible ($M$-i. d.) if there exists a sequence of integers $0<n_1<n_2<\dots$ such that for every $n_j$ it equal to the distribution of the product of some $n_j$ independent identically distributed random variables.
It is shown that the stable laws with parameters ($\alpha<1$, $\gamma=0$), ($\alpha=1$, $\beta=0$), ($\alpha>1$, $\gamma=0$, $\beta=0$) are $M$-i. d. but the stable laws with parameters ($\alpha>1$, $\gamma=0$, $\beta\ne0$) are not $M$-i. d.