Thus, let $\mathrm{OPP}$ be the axiom that $|A|\lt|B| \Rightarrow |2^A|\lt|2^B|$ for any sets $A$ and $B$; and, for any ordinal $\alpha$, let $\mathrm{CH}_\alpha$ be the hypothesis that $\aleph_\alpha=\frak c$ (so that $\mathrm{CH}_1=\mathrm{CH}$) . Define $S$ to be the set of those ordinals $\alpha\in\frak c$ such that $\mathrm{CH}_\alpha$ does not provably (within $\mathrm{ZFC}$) violate $\mathrm{ZFC}$ (for example, it is known that $\omega\backslash${$0$}$\subseteq S$ and $\omega \notin S$); and let $S'$ be the set of those $\alpha\in\frak c$ such that $\mathrm{CH}_\alpha$ does not provably (within $\mathrm{ZFC}$) violate $\mathrm{ZFC}$ $\&$ $\mathrm{OPP}$. Clearly $S'\subseteq S$. But is $S'= S$ ? Or are any elements of $S$ known to be not in $S'$ ?

My guess is that $\mathrm{OPP}$ can't restrict the possibilities for violations of $\mathrm{CH}$ because the sets it talks about in the consequent---especially $2^B$--- are too big to be relevant; but I'm not sure of my footing here.