I am still unable to fix the mistake in the paper in the previous post.

The method in that paper ends up with a “limit model of $\mathrm{HOD}^{L[x]}$” whose Woodin cardinal is $u_{\omega}$. Therefore, this “limit model” up to its Woodin is “obviously” $L_{u_\omega}[\mathcal{O}_{\Sigma^1_3}]$ (cf. Iterates of $M_1$), but I don’t see why. Maybe it’s just as hard as computing $\mathrm{HOD}^{L[x]}$.