Let $L$ be a restricted Lie algebra over a field $F$ of characteristic $p>0$. Assume that the following condition holds:
For every restricted ideal $I$ of $L$, the minimal restricted subalgebras of $L/I$ are pairwise non-isomorphic.
QUESTION: Is $L$ necessarily abelian?
I already know that the answer is affirmative if one assumes that $L$ is nilpotent.