We say that a Turing degree $a$ is a strong minimal cover of $b$ if $a$ is strictly above $b$ and if any $c$ strictly below $a$ is (not necessarily strictly) below $b$. It is known that some degrees do have strong minimal covers and some other don't.

It is known that 0 has strong minimal covers. Those strong minimal covers are called minimal degrees. It is (as far as I know) an open question to know whether every minimal degree has itself a strong minimal cover.

My question is : can the question about minimal degree be extended to every degree which is a strong minimal cover of some other degree ? or is it known (for obvious or complicated reasons) that there is some strong minimal covers which don't have themselves strong minimal cover ?