Graded-ring conjecture for absorbing ideals

Let $R=\bigoplus_{g\in G}R_g$ be a ring graded by a torsion-free abelian group $G$. Let $\alpha$ be an ideal of $R$, and let $\alpha^*$ be the largest homogeneous ideal contained in $\alpha$. For an ideal $\beta$, let $\omega(\beta)$ denote its absorbing number.

Graded-ring conjecture. For any ideal $\alpha$ of a graded ring $R$,

$$\omega(\alpha^*)\leq\omega(\alpha).$$

This is motivated by the fact that the homogeneous part of a prime ideal is prime, and asks whether the analogous inequality holds for absorbing ideals. The source does not state that the conjecture has been resolved.