One of the unexplained phenomena about large cardinal axioms is that they tend to be in a linear order (in terms of consistency strength). So it brings to mind other natural questions about this "order".
So my question is - For any two large cardinal axioms $A1$, $A2$ such that $ZFC + A1$ proves $ZFC + A2$ is consistent, can we define a large cardinal axiom $A3$ such that $ZFC + A1$ proves $ZFC + A3$ is consistent, and $ZFC + A3$ proves $ZFC + A2$ is consistent?
Meaning, is the order known\thought to be densely ordered (assuming the ordering itself is not some random phenomena)? Alternatively, can we build an example (even a highly contrived one) of $A1$, $A2$, where any axiom we add to $ZFC$ is either stronger than $ZFC + A1$, weaker than $ZFC + A2$, or equiconsistent to one of them (meaning the ordering is not dense)?
P.S - I know there is no agreed upon definition of "large cardinal axiom", but if we discuss their linear order we should be able to ask questions like this as well.