The static hypothesis seems to rely on common sense rather than hard evidence. It's rare for somebody who seems dull to become a genius all of a sudden. And when it does happen it's easy to assume they seemed dull rather than actually were.
There is no disconnect between intelligence being simple yet hard to improve. Losing weight is very simple. But it's hard. Obesity is only slightly less heritable than IQ (both highly heritable), yet we know for a fact it's 100% controllable by the individual (unless one is strongly against the notion of free will).
Free will has nothing to do with not being able to do something because physical constraints are placed on a person. e.g. A person in a wheel chair does not have more or less free will then someone who can walk. Even though the latter clearly has more freedom in a certain sense.
Alternatively, one could claim many simply don't want to lose weight. Then it should be fair game to claim many don't want to improve their intelligence.
Dual n back studies are a good starting point. Could go through gwern's article [0]. Which I very much disagree with, but it does cover a lot of research. He essentially concludes the active placebo studies that produce as many gains as dnb prove dnb doesn't work. But without ground truth there's no way to know these active placebo gains are fake. Many of these active placebos are other cognitive training methods, it's perfectly plausible they may not be placebos at all. IMO we can only conclude that if dnb does work it isn't uniquely great.
Another thing to remember - if we want to determine if it is possible to improve intelligence we should care about maximum gains not average gains. Again, without ground truth, one cannot simply claim big gainers are meaningless outliers that can be discounted. There may not be anything producing 2 SD on average (though plenty showing much more than the 3 points you mention), but many such improvements have been recorded.
Imagine weight was something we couldn't observe or understand. Based on statistical science, many may similarly conclude it is impossible to change it.
That's way too much black and white thinking. It's pretty safe to say most obese people would like to lose weight. They just don't want to sacrifice the things they are doing, either consciously or subconsciously. This is orthogonal to free will. What you are talking about is the whether the conscious mind can win over the subconscious mind. That's called discipline. Not free will.
So if you've led a non ideal lifestyle in the past, you might be able to increase your intelligence by making less mistakes, relative to your current abilities.
If measured as IQ, the variance tends to be around 5 points on most tests.