I don't see how you could prove either way. The switch with halo 2 is less than 1 second.
Did you read the post I linked? I can't determine the exact extent to which Classic Mode isn't active during Anniversary Mode, but it's clear that not everything is running. There's also no obvious performance crossover between modes, although as noted I wasn't really able to test H2C load in H2A since H2C is so stable (all other conditions for both oXbox games are testable with negative results, including CE performance not affecting CEA performance).
(As to why they'd stretch the truth, resolutiongate would be a reason. It seems silly, but this is the same studio behind crap like Fourteen Day Buy and Play.)
When you switch back and forth there is no texture or geometry pop in whatsoever.
As I noted earlier in the thread, it's probable that both audio layers are running all the time, and Classic Mode will have to have a RAM allocation in addition to any OG memory management (i.e. streaming) requirements. Although the general lack of streaming-related pop-in makes me wonder if that's even an issue (or if they just load in everything at the start).
Halo 2 anniversary has slightly better graphics then Halo 4 in the MCC.
In some ways it's doing
a lot more than Halo 4 is. Dynamic lights are often much larger, have specularity, and are capable of being spotlights. Some "static" lights are handled as their own precise sources, not just as part of an environmental lightmap. Some materials seem to have more expressive shading than in Halo 4, especially glossy metalish stuff. Screen-space reflections.
Much higher-quality reflections for special/transparent surfaces like water (responds to lighting, sometimes reflects level geometry i.e. through what seems like might be box-projected maps, etc). Transparencies don't seem to have ordering issues.
I'm not very fond of how H2A looks, but regardless of the extent to which H2C is active in the background, it wouldn't shock me if it could merit resolution compromise in circumstances where Halo 4 didn't. Remember that on 360, CEA had lower resolution than Halo 4 (and in that case it definitely wasn't running both layers simultaneously), and in some ways H2A is more intensive than CEA.