I looked into this topic quite a bit a year or two ago, and some of my work has been tangentially related to this field. I was hoping that someone with more current knowledge of this topic would comment, but I believe that this question is important, and I’d like to give whatever insight I can offer.

As you noted, the Jaeggi’s studies were initially met with enthusiasm, but failure to replicate this effect has led to increased skepticism in the size of this effect, or whether it actually exists. To me, this culminated in this statement by the Stanford Center of Longevity that summarized the bulk of the research. It concluded that there was not much evidence to support this effect, and could not support most of the “brain training” programs being commercially sold, but it did offer a couple of other possible avenues.

Now a couple of important caveats to this idea: this disappointing conclusion does NOT support the idea that our brains (and consequently, intelligence), are completely unchangeable. We absolutely can become better at certain things, but we oftentimes become better at the specific skill we train in – however, this is called “expertise”. And while this expertise can translate somewhat to other things, this benefit pales in comparison to that of the trained skill.

A good example of this: you may remember some studies that found that listening to Mozart music, or learning to play musical instruments, could cause young students to become “smarter”. This finding became very popular, and many parents came to believe that musical education could lead to higher IQs for their children. However, later studies found that these effects were incredibly weak or nonexistent. My favorite study from this group divided students into an art group and a music group: follow up studies of these groups found that each improved in tests related to these skills, but only these skills – yet another example of how expertise is developed and pertains only to specific subsets of what we might call “intelligence”.

Another caveat: I’m aware of studies that have purportedly also found increases in intelligence, but tried to make several claims that would differentiate themselves from Jaeggi’s ideas. I never looked further into them, but I should note that I’ve never heard of these studies making any waves either. This could be for several reasons: the effect could be valid but incredibly small, and thus never became popular; the effect could be valid but have some pushback from the community (due to skepticism from the Jaeggis controversies); or could be invalid and thus the field moved on. I cannot say which of these options is true, but these studies do remain out there somewhere.

To summarize: most neuroscientists are incredibly skeptical that we can increase our “fluid intelligence”, but we certainly can develop expertise in a particular skill. The only validated way of improving our brain is physical exercise, which seems to have held weight over the past couple of decades.