Suppose that $\cal P$ is a class of forcings, we denote by $\operatorname{FA}_\kappa(\cal P)$ the statement that whenever $\Bbb P\in\cal P$, $\gamma<\kappa$, and $\{D_\alpha\mid\alpha<\gamma\}$ are dense open sets in $\Bbb P$, then there is a filter $G\subseteq\Bbb P$ such that $G\cap D_\alpha\neq\varnothing$ for all $\alpha<\gamma$.

The usual examples are Martin's Axiom where $\kappa=2^{\aleph_0}$, and $\cal P$ is the class of ccc forcings; or the Proper Forcing Axiom where $\cal P$ is the class of proper forcings and $\kappa=2^{\aleph_0}$ (which then implies $\kappa=\aleph_2$).

Let's focus on subclasses of ccc forcings.

We can force Martin's Axiom for only $\sigma$-centered forcings, or other subclasses, and there is a clear hierarchy here. For example, $$\operatorname{FA}_{2^{\aleph_0}}(\textrm{ccc})\implies\operatorname{FA}_{2^{\aleph_0}}(\sigma\textrm{-centered})\implies\operatorname{FA}_{2^{\aleph_0}}(\sigma\textrm{-linked}).$$

Of course, there are many, many other subclasses that one can consider.

Question.Is there any work on separating the various forcing axioms for subclasses of ccc forcings? For example, is it consistent that $\operatorname{FA}_{\aleph_2}(\sigma\textrm{-linked})$ holds, but only $\operatorname{FA}_{\aleph_1}(\sigma\textrm{-centered})$ holds, and $\operatorname{FA}_{\aleph_1}(\textrm{ccc})$ fails altogether?

(And I cannot stress this enough, the choice of subclasses was entirely arbitrary, and any other subclasses of ccc forcings are of interest here.)