May I ask you which is exactly your (preferred) subfield of work?

My research was in a sense Abbott-like: how a multi-dimensional world would look to someone living in the lower dimensions. It is different from the standard string-theoretical approach of bulk-vs-brain, because it is non-perturbative. I can certainly go into the details of it, but probably not in this comment.

What are the most important open problems in that field that you think could receive decisive insight (both theoretically and experimentally) in the next 10 years?

Caveat: I'm not in academia at this point, so take this with a grain of salt.

Dark energy (not to be confused with Dark matter) is a major outstanding theoretical problem in GR. As it happens, it is also an ultimate existential risk, because it limits the amount of matter available to humanity to "only" a few galaxies, due to the accelerating expansion of the universe. The current puzzle is not that dark energy exists, but why there is so little of it. A model that explains dark energy and makes new predictions might even earn the first ever Nobel prize in theoretical GR, if such predictions are validated.

That the expansion of the universe is accelerating is a relatively new discovery (1998), so there is a non-negligible chance that there will be new insights into the issue on a time frame of decades, rather than, say, centuries.

In observations/experiments, it is likely that gravitational waves will be finally detected. There is also a chance that Hawking radiation will be detected in a laboratory setting from dumb holes or other black-hole analogs.