Restricted notions of set-theoretic geology We say that $W$ is a ground of the universe $V$ if $W$ is a model of ZFC and there is a poset $P\in W$ such that $W[G]=V$ for some $G$ which is $P$-generic over $W$. The Ground Axiom ($\text{GA}$) asserts that $V$ has no nontrivial grounds while $\text{DDG}$ is the statement: for all grounds $W_1$, $W_2$ of $V$ there is a ground $U$ (of $V$) contained in $W_1\cap W_2$.
It has already been proven that $\text{GA}$ can be forced (by means of a class forcing notion) and that $\text{DDG}$ is a theorem of ZFC. I'm interested in restricting these notions. For example, $\text{GA}_{\text{$\sigma$-closed}}$ is the assertion that the universe is not a set-forcing extension of an inner model $\sigma$-closed forcing notion.
Similarly $\text{DDG}_{\text{$\sigma$-closed}}$ is the statement: for all $\sigma$-closed grounds $W_1$, $W_2$ of $V$ there is a $\sigma$-closed ground $U$ (of $V$) contained in $W_1\cap W_2$. Of course, $\text{GA}\rightarrow \text{GA}_{\text{$\sigma$-closed}}$. How about the converse? That is, are there models of ZFC satisfying $\neg\text{GA}+\text{GA}_{\text{$\sigma$-closed}}$ or $\neg\text{GA}+\text{GA}_{\text{ccc}}$? This question arises from Reitz - The ground axiom and when it was published the question was still open.
Moreover, I'm not able to exhibit models in which

*

*$\text{DDG}_{\text{$\sigma$-closed}}$ fails,

*$\text{DDG}_{\text{ccc}}$ fails.

The last two questions are probably easier.
 A: If you add a Cohen real to $L$ (or any model of the Ground Axiom), you get a model of the countably closed Ground Axiom, since all intermediate extensions are grounds for Cohen forcing. On the other hand, if you add a Sacks real to $L$, you obtain a model with no ccc grounds since Sacks forcing is not ccc and there are no intermediate extensions. So the Sacks extension is a model of the ccc Ground Axiom but not the Ground Axiom.
A counterexample to the countably closed DDG can be obtained by adding mutual generics $G$ and $H$ for $\text{Col}(\omega_1,\mathbb R)$ to $L(\mathbb R)$ when $L(\mathbb R)$ is a model of DC but not AC. Then $L(\mathbb R)[G\times H]$ has $L(\mathbb R)[G]$ and $L(\mathbb R)[H]$ as countably closed grounds, but their intersection is $L(\mathbb R)$, and moreover any countably closed ground of the two models would contain all the reals, and hence be $L(\mathbb R)$. Since AC fails in $L(\mathbb R)$, it doesn't count as a ground. (Edit: Actually I am remembering now that this example was pointed out by Gunter Fuchs at the inner model theory meeting in Girona in 2018 after I had gone overboard using $\mathbb P_\text{max}$.)
To make Asaf happy, if possible, I also point out that assuming ZFC, the countably distributive DDG is true for ZF grounds, though the example above shows it fails for ZFC grounds. (Edit: I mean the intersection of any two ZF grounds closed under countable sequences contains a ZF ground closed under countable sequences. But this need not be a ground of the ZF grounds we started with.) If $M_0$ and $M_1$ are ZF grounds, there is a ZFC ground $N$ contained in both of them by a result of Usuba. Thus $N(\text{Ord}^\omega)$ is a common inner model of $M_0$ and $M_1$ that is closed under $\omega$-sequences. Now to show that $N(\text{Ord}^\omega)$ is a ground, it suffices by a theorem of Grigorieff to show $N(\text{Ord}^\omega) = N(X)$ for some set $X$. Let $\delta$ be large enough that $N$ has the $\delta$-cover property, and we claim $N(\text{Ord}^\omega) \subseteq N(\delta^\omega)$. To see this, suppose $\sigma$ is a countable set of ordinals. Then there is a set of ordinals $\tau\in N$ covering $\sigma$ of cardinality less than $\delta$. Let $f : \gamma\to \tau$ be the increasing enumeration of $\tau$. Then $\gamma < \delta$, so $\bar \sigma = f^{-1}[\sigma]\in N(\delta^\omega)$. Therefore $\sigma = f[\bar \sigma]\in N(\delta^\omega)$.
I couldn't figure out a counterexample to ccc DDG, though, but I don't doubt that there is one.
