Probably still open. James Barnes' dissertation (2018) addresses initial segments under the arithmetic reducibility, but is not specifically about $\omega$-CEA degrees.
Barnes, James S., On the decidability of the $\Sigma_2$ theories of the arithmetic and hyperarithmetic degrees as uppersemilattices, J. Symb. Log. 82, No. 4, 1496-1518 (2017). ZBL1391.03031.