Counterexamples to the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis This question of mine comes from Woodin's "In Search of Ultimate-L" article, where they define the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis. My confusion arises from remark $4.15$ and theorems $4.16$ and $4.17$ on pages $32$-$33$:
To quote the $3$-line paragraph directly above theorem $4.16$:

We give two counterexamples to the attempt of formulating variations
of the iteration hypotheses above by weakening the requirement that the
iteration trees be $0$-strongly closed and maximal.

Now I read this in two different ways according to the theorems that come next:
First, I can take this to mean that both the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis are wrong in the presence of a supercompact cardinal and we can build counterexamples which are even strongly maximal, let alone maximal.
Second, I can take this to mean that there has been a typo in the theorems or the definitions considered here, and the definitions of the Weak Unique Branch Hypothesis and Weak $(\omega_1+1)$-Iteration Hypothesis should have been stated for $0$-strongly closed strongly maximal iteration trees and the theorems should have given us just $0$-strongly closed maximal iteration trees. And I have a bit of evidence supporting some sort of a typo occurring, because in "Suitable Extender Models I", the same counterexample theorem produces totally non-overlapping trees, but you may take this with a grain of salt since I don't know whether totally non-overlapping is equivalent to strongly maximal or not.
All in all, I would appreciate it if anyone could tell me what the correct case is and what is being implied here.

EDIT I: I am adding the relevant definitions and theorems for the sake of completeness, since I think some of them might not be super standard or well-known:

Definition $4.13$ (Weak $(\omega_1+1)$-Iteration Hypothesis). Suppose that $(M, \delta)$ is a countable course premouse and $\pi: M\rightarrow V_\Theta$ is an elementary embedding. Then $(M, \delta)$ has an iteration strategy of order $(\omega_1+1)$ for $0$-strongly closed maximal iteration trees on $(M, \delta)$.


Definition $4.14$ (Weak Unique Branch Hypothesis). Suppose that $(V_\Theta, \delta)$ is a countable course premouse and that $\mathcal{T}$ is a countable $0$-strongly closed maximal iteration tree on $(V_\Theta, \delta)$ of limit length. Then $\mathcal{T}$ has at most one cofinal wellfounded branch.


Theorem $4.16$. Suppose there exists a supercompact cardinal. Then there exists an extender $E$ such that $\nu_E = (2^{2^{\kappa}})^{M_E}$, where $\kappa = \kappa_E$ and $M_E = \text{Ult}_0(V;E)$, and a $0$-strongly closed strongly maximal iteration tree $\mathcal{T}$ of length $\omega$ such that $\mathcal{T}$ has exactly two wellfounded cofinal branches.


Theorem $4.17$. Suppose there exists a supercompact cardinal. Then there exists an extender $E$ such that $\nu_E = (2^{2^{\kappa}})^{M_E}$, where $\kappa = \kappa_E$ and $M_E = \text{Ult}_0(V;E)$, and a $0$-strongly closed strongly maximal iteration tree $\mathcal{T}$ of length $\omega^2$ such that $\mathcal{T}$ has only one cofinal branch and that branch is not wellfounded.

Remarks:

*

*First, I want to mention that the bold parts in the texts are the maximality conditions I was talking about in the main bulk of the post,


*and second, I want to mention that the theorems $4.16$ and $4.17$, that I have listed here, are weaker than what is written in the article, but they suffice for the purpose of my question.
 A: I think the main issue is that the trees $\mathcal{T}$ referred to in Theorems 4.16, 4.17,
are not on $V$, but on $\mathrm{Ult}(V,E)$. In both cases we can form a tree $\mathcal{U}$ on $V$ given by $E$ followed by $\mathcal{T}$, but then $\mathcal{U}$ is not strongly closed (as it is not true that $\mathrm{strength}(E)=\mathrm{lh}(E)=$ an inaccessible cardinal (in $V$)).
However, I don't see how Theorem 4.17 itself violates  the strengthening of the Weak $(\omega_1+1)$-Iteration Hypothesis, because (i) 4.17 doesn't say that the branches at the intermediate limit stages are unique wellfounded branches, and (ii) the tree is on $V$, not a countable elementary substructure of some $(V_\theta,\delta)$. (It does directly contradict  CBH, the Cofinal Branches Hypothesis.) There is related material in Neeman-Steel "Counterexamples to the unique and cofinal branches hypotheses". In the tree there which is a counterexample to CBH (the tree $\mathcal{U}$ constructed at the end), the tree has length $\omega^2$, and consists of a single extender followed by an $\omega$-sequence of alternating chains, hence at each intermediate limit stage $n\times\omega$, there are exactly 2 cofinal branches. But in their construction, they say both of these branches give wellfounded models (in fact they are the same model). So it is not clear to me that the tree didn't just follow the wrong iteration strategy. Secondly, suppose that the tree $\mathcal{T}$ referred to in 4.17 does have unique wellfounded branches at intermediate stages. Then one might try to form a countable elementary hull $M$ of some $(V_\Theta,\delta)$ so that the picture reflects down to $M$. This would work fine if e.g. the intermediate stages are all alternating chains. But if they are instead complex enough that they have many cofinal branches, then uniqueness of wellfounded branches doesn't seem to automatically pass down to $M$. So as far as I can tell, this also needs to be dealt with.
