It's known that CH is equivalent to existence of unique $\eta_1$-set of size continuum.
Def: Call an ordered set $(X, <)$ an $\eta_1$-set if for countable $A, B\subseteq X$ with $a < b$ for all $a\in A, b \in B$ there exist $v\in X$ with $a < v < b$ for all $a\in A, b\in B$.
All $\eta_1$-sets are of size at least continuum, and the smallest size of one is always continuum.
It's known that there exists an $\eta_1$-set $\mathbf{Q}$ such that $\mathbf{Q}$ embeds into every $\eta_1$-set. This $\eta_1$-set doesn't contain a copy of $\omega_2$.
If CH is false, then $\mathbf{Q}$ and $\omega_2\times\mathbf{Q}$ with lexicographic order are non-isomorphic $\eta_1$-sets of size continuum.
Note that there is at most $2^\mathfrak{c}$ structures of $\eta_1$-set on a set of size continuum. Is there precisely that many non-isomorphic $\eta_1$-sets of size continuum when CH is false?
Say that a totally ordered set is a weak $\eta_1$-set if it satisfies the $\eta_1$ property restricted to pairs $(A,B)$ such that $A$ has no greatest element and $B$ has no least element. If $X$ is a weak $\eta_1$-set and $Y$ is an $\eta_1$-set, then $X\times Y$ with the lexicographic order is an $\eta_1$ set (the fact that $Y$ is an $\eta_1$-set means you don't have to worry about the case where the first coordinates in $A$ or $B$ are constant). Moreover, in the special case where $Y=\omega_2\times\mathbf{Q}$, you can recover $X$ up to isomorphism from $X\times Y$, as the equivalence classes in $X\times Y$ where you say $a\sim b$ if there is no copy of $\omega_2$ between $a$ and $b$.
So, it suffices to show there are $2^{\mathfrak{c}}$ different isomorphism classes of weak $\eta_1$-sets of size $\mathfrak{c}$. To prove this, take an arbitrary subset $A$ of the ordinal $\mathfrak{c}$ and replace each element of $A$ with a copy of $\mathbf{Q}$. This will always result in a weak $\eta_1$-set $X_A$. Moreover, if you restrict $A$ to consist of only limit ordinals, then you can recover $A$ from the isomorphism class of $X_A$, since the copies of $\mathbf{Q}$ in $X_A$ can be identified as the maximal intervals with no greatest or least element that are densely ordered. Since there are $2^{\mathfrak{c}}$ different such choices of $A$, this gives $2^{\mathfrak{c}}$ isomorphism classes of weak $\eta_1$-sets of size continuum.