GCH implies acceptability

Question

I have been studying the concept of acceptability, particularly in its relation to GCH. There are many versions of it in the sources I have found, with some slight variations, and some of them are claimed to be equivalent to GCH, some of them are claimed not to be equivalent to GCH. For example, Welch, in A Condensed History of Condensation, claims that GCH is equivalent to acceptability but not to weak acceptability. Friedman and Holy, in A Quasi-Lower Bound on the Consistency Strength of PFA, define a slightly different form of weak acceptability and prove that it is equivalent to GCH. Schindler and Zeman, in Fine Structure, define a fine-structural version of acceptability and say that it can be seen as a stronger form of GCH.

I am interested in any further references providing details and clarifying the relation between acceptability and GCH.
I am particularly interested in knowing whether the fine-structural version given in Schindler and Zeman is implied by GCH in some interesting class of models. More specifically, in the models $\textbf{L}[A]$, does GCH imply that for some appropriate choice of $A$ the $J$-structures $J^A_{\alpha}$ are acceptable for some unbounded class of $\alpha$'s in the sense given in Fine Structure Theory?

EDIT

A more basic question may be helpful. In which precise sense, if any, is fine-structural acceptability (defined for the $J$-hierarchy) equivalent to acceptability defined directly for the $L$-hierarchy?

Answer 1

Another related reference, is the following “An Abstract Condensation Property ” by David Richard Law.

Here is the abstract of it:

Let $A = (A, \dotsc)$ be a relational structure. Say that $A$ has condensation if there is an $F : A^{< \omega} → A$ such that for every partial order $P$, it is forced by $P$ that substructures of $A$ which are closed under $F$ are isomorphic to elements of the ground model. Condensation holds if every structure in $V$, the universe of sets, has condensation. This property, isolated by Woodin, captures part of the content of the condensation lemmas for $L$, $K$ and other "$L$-like" models. We present a variety of results having to do with condensation in this abstract sense. Section 1 establishes the absoluteness of condensation and some of its consequences. In particular, we show that if condensation holds in $M$, then $M \models \mathrm{GCH}$ and there are no measurable cardinals or precipitous ideals in $M$. The results of this section are due to Woodin. Section 2 contains a proof that condensation implies $\Diamond_κ(E)$ for $\kappa$ regular and $E \subseteq \kappa$ stationary. This is the main result of this thesis. The argument provides a new proof of the key lemma giving GCH. Section 2 also contains some information about the relationship between condensation and strengthenings of diamond. Section 3 contains partial results having to do with forcing "$\operatorname{Cond}(A)$", some further discussion of the relation between condensation and combinatorial principles which hold in $L$, and an argument that $\operatorname{Cond}(G)$ fails in $V[G]$, where $G$ is generic for the partial order adding $\omega_2$ cohen subsets of $\omega_1$.