In his book "The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal" Woodin formulated tho following problem [No 22]:
Are there two $\Pi_2$-sentences $\psi_1$ and $\psi_2$ in the language of the structure $(H(\aleph_2), \in, \omega_1, NS_{\omega_1})$ s.t. each $\psi_1$ and $\psi_2$ is $\Omega$-consistent with CH but s.t. $\psi_1 \wedge \psi_2$ $\Omega$-implies $\neg CH$.
(Here just think of "$\Omega$-consistent" as something weaker than "provably forceable from large cardinals" and of "$\Omega$-implies" as something weaker than just "implies".)
The answer to this question is no, as Aspero, Larson and Moore have shown in "Forcing Axioms and the CH".
I am curious about what consequences have been received since then from this result.