# Downward density of w-REA sets under arithmetic reducibility?

Is the question of the downward density of the w-REA sets under $$\leq_a$$ still open? If not can anyone point me to a proof? That is do we know if for every $$\omega$$-REA set $$X >_a 0_a$$ there exists another $$\omega$$-REA $$Y$$ with $$X >_a Y >_a 0$$?

I presume it is still open and it's an interesting question IMO but before wasting any time on it I figured I'd ask a bit more widely to be sure.

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.