What is HYP? The minimal model?

Let us continue this discussion in chat.

It's Cartesian product. I'm splitting $X$ into two interleaved subsequences, which are in turn split into $|X|$ many $\omega$-sequences.

The variant in the linked post is slightly different. Rather than there being one box for every set of reals (i.e. the index set is $\mathcal{P}(\mathbb{R})$), I have the index set $X$ be the Hartogs number of $\mathbb{R}.$ I was just describing how to construct this set.

Is it strong enough to rule out the variant I describe in the second paragraph? My answer isn't about the standard 100 mathematician puzzle.

@ZuhairAl-Johar That's right.

@ZuhairAl-Johar Iterated power set.

@DouglasUlrich That's a proper class in this model.

@ZuhairAl-Johar Integers.

And here I was debating whether to include AC$_{WO}$ in my list of choice principles at the beginning. Good to see this axiom put to use.

$X$ is the family of sets I want a choice function on, not $P(X).$

I mean set-generic but an answer for class-generic would be nice as well.

I am including trivial forcing. I don't see how that gives a positive answer. I'm only assuming the generic extensions (including $V$) satisfy weak choice principles.