I believe it is still open whether $\mathrm{NS}_{\omega_2} \restriction \mathrm{cof}(\omega_1)$ can be saturated. But it was known by early unpublished work of Woodin that $\mathrm{NS}_{\omega_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega_1)$. The complement provides a "safe space" to allow the iteration to work. Details are given by Foreman-Komjath (MR: 2151585) and by me.

I believe it is still open whether $\mathrm{NS}_{\omega_2} \restriction \mathrm{cof}(\omega_1)$ can be saturated. But it was known by early unpublished work of Woodin that $\mathrm{NS}_{\omega_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega_1)$. The complement provides a "safe space" to allow the iteration to work. Details are given by Foreman-Komjath (MR: 2151585) and by me.

I have heard rumors that recent work related to higher analogues of forcing axioms might be used to tackle the problem. This approach would surely give a model with not-GCH, so we'd have an obvious next open question to tackle.