Probably still open. James Barnes' dissertation (2018) addresses initial segments under the arithmetic reducibility, but is not specifically about $\omega$-CEA degrees. 

<cite authors="Barnes, James S.">_Barnes, James S._, [**On the decidability of the $\Sigma_2$ theories of the arithmetic and hyperarithmetic degrees as uppersemilattices**](http://dx.doi.org/10.1017/jsl.2017.51), J. Symb. Log. 82, No. 4, 1496-1518 (2017). [ZBL1391.03031](https://zbmath.org/?q=an:1391.03031).</cite>