A tree $T$ on $\omega \times \lambda$ is weakly homogeneous if there is a countable set $\sigma$ of countably complete measures on ${}^{<\omega}\lambda$ so that $x \in p[T]$ if and only if there is a countably complete tower of measures $\bar{\mu} = \langle \mu_i : i \in \omega \rangle$ so that each $T_{x\upharpoonright i} \in \mu_i$ and each $\mu_i \in \sigma$.

Fix such a tree $T$ and a $\sigma$ witnessing weak homogeneity. Fix an enumeration $\langle \mu_i : i \in \omega\rangle$ of $\sigma$ so that projections of $\mu_i$ come before $\mu_i$.

For an ordinal $\nu$, the Martin-Solovay tree $\mathrm{MS}_\nu(T,\sigma)$ is a tree on $\omega \times \nu$ defined by: $(h,s) \in \mathrm{MS}_\nu(T,\sigma)$ if and only if for all $i < l < |s|$, if $k_i = \dim(\mu_i)$ and $k_l = \dim(\mu_l)$, $T_{s \upharpoonright k_i} \in \mu_i$, $T_{s \upharpoonright k_l} \in \mu_l$, and $\mu_i$ is a projection of $\mu_l$, then $j_{i,l}(h(i)) > h(l)$, where $j_{i,l}$ is the natural map between the $\mu_i$ ultrapower and the $\mu_l$ ultrapower.

In Steel's "The Derived Model Theorem" Lemma 1.19, he shows that $p[T] = {}^\omega\omega \setminus p[MS_{\lambda^+}(T,\mu)]$. This is proved by finding a continuous witness to the ill foundedness of all appropriate towers coming from $\sigma$. At least in the Steel's proof provided there, it seems that he is using the axiom of choice.

So my question is: Under $\mathsf{AD}_\mathbb{R}$, Martin showed that every tree $T$ on $\omega \times \lambda$ where $\lambda < \Theta$ is weakly homogeneous. Fix $T$ and $\sigma$ witnessing weak homogeneity. The Martin-Solovay tree $\mathrm{MS}_{\lambda^+}(T,\sigma)$ can be constructed. Under $\mathsf{AD}_\mathbb{R}$, is $p[T] = {}^\omega\omega \setminus p[MS_{\lambda^+}(T,\sigma)]$ still true?

In Larson's Stationary tower book, he mentioned that it is unknown whether the type of continuous witnesses to illfoundedness of all towers (used in $\mathsf{AC}$ proof) can be proved in $\mathsf{ZF + DC}$.

In his Games and Scales paper, Steel mentioned that Martin-Solovay trees are studied in the choiceless $\mathsf{AD}$ context. I suspect at least under determinacy the Martin-Solovay tree should project onto the complement of a weakly homogeneous tree.

Thanks for any information or references to this question.