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 $L$ is nilpotent, or $L$ is finite-dimensional and $F$ is algebraically closed.