Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$. Then does $V[G]$ satisfy the following?
For a generic ultrafilter $H\subseteq \mathrm{NS}^+_{\omega_1}$ over $V[G]$ and a function $f\colon [\omega_1]^{<\omega}\to 2$ in $V[G]$, there is a set $S\in H$ such that $f$ is homogeneous over $S$.
Or can we force the above statement from $\mathsf{ZFC}$ with some large cardinal axiom?
The motivation for this question is whether the following variant of Prikry forcing can have the Prikry property:
$\mathbb{P}$ is the set of all $(a,S)$ where $a$ is an finite sequence over $\omega_1$ and $S$ is a stationary subset of $\omega_1$. $(b,T)\le (a,S)$ if $b$ is an end extension of $a$, $T\subseteq S$, and $b\setminus a\subseteq S$.
This forcing may not preserve cardinals larger than $\omega_2$ (the usual argument for the Prikry forcing over a measurable $\kappa$ is $\kappa$-c.c. does not work due to the extra condition $T\subseteq S$ in the definition of $\le$), but it seems that the above Ramsey-like property for $\mathrm{NS}^+_{\omega_1}$ guarantees $\mathbb{P}$ has a version of Prikry property, so $\mathbb{P}$ does not add bounded countable subsets of $\omega_1$.
I am more interested in the case when $\omega_1$ is replaced by another regular cardinal, and the most interesting case for me is $\omega_{\omega+1}$, which might give another way to singularize $\omega_{\omega+1}$ with adding no bounded subsets of $\omega_{\omega+1}$.
