But, alas - "twin K got a higher test score than twin A, after twin K went to test-prep school" is not a headline to get many clicks.
Just like you, I think IQ is very trainable. It is very common to have questions about logic, which may be represented as Venn Diagrams, or visual questions with multiple parts, each one following a logic...
<appeal to authority> I say that as someone who "started" with 110 QI according to a Orkut online test, and grinded my way to 145 according to different online tests. </appeal to authority>
For that reason people who learned how to prove things in math are naturally going to have a higher IQ: they learned how to solve some questions in the test.
“Psychometrics is a soft science with poor replicability except IQ which is as diamond hard as physics and is reliable enough to make broad generalizations about people and explain away just about any difference in outcome.” - the HBD or whatever it’s called this week crowd.