The internal structure of ${\mathrm{H}\mathrm{O}\mathrm{D}}^{L[x]}$ up to its Woodin

Assume ${\mathbf{\Delta }}_3^1$-determinacy. It is shown that for any $x \geq_T M_1^{#}$,$$\mathrm{H}\mathrm{O}\mathrm{D}$^{L[x]}$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to ${\omega }_2^{L[x]}$.