This follows up on Incomparable dense linear orderings extending $\langle \mathbb{R},< \rangle$ and hopefully sparks more discussion.

Where $a<b$, say that the four “types” of nonempty bounded intervals are:

$(a,b)$, $[a,b]$, $(a,b]$, and $[a,b)$.

Let $\langle X,< \rangle$ and $\langle Y,< \rangle$ be dense linear orderings without endpoints such that

$\langle \mathbb{R},< \rangle$ is order-embeddable into $\langle X,< \rangle$, and into $\langle Y,< \rangle$,

all the nonempty bounded intervals of $\langle X,< \rangle$ of the same type are order-isomorphic,

all the nonempty bounded intervals of $\langle Y,< \rangle$ of the same type are order-isomorphic,

$\langle X,< \rangle$ is isomorphic to its inverse order, and

$\langle Y,< \rangle$ is isomorphic to its inverse order.

Are there examples of $\langle X,< \rangle$ and $\langle Y,< \rangle$ such that

“no nonempty open interval of $\langle X,< \rangle$ is order-embeddable into any nonempty open interval of $\langle Y,< \rangle$, and no nonempty open interval of $\langle Y,< \rangle$ is order-embeddable into any nonempty open interval of $\langle X,< \rangle$”?

Is it independent of $ZFC$ whether there are such pairs of orderings for which the statement in quotes holds?

If $ZFC$ proves that there are such pairs of orderings, then which definitions (if any) of such orderings $\langle X,< \rangle$ and $\langle Y,< \rangle$ are such that it is independent of $ZFC$ whether the statement in quotes holds of them?