I'm usually the first to think the H is lying in these cases. In your case, I'm not so inclined -- it seems just as likely to me that your gut feeling is caused by projecting your own feelings around your own lack of interest in the M, intimacy issues, (discussed on your other thread) and EA onto H. It almost seemed to me that you came here so that you could quit looking at things about yourself that are (very understandably) difficult to examine and try to change.
BUT that is just me. AND I also know that the gut is very nearly always exactly right on this stuff.
So, I guess I have no grip on what to think about H re the extracurriculars in your case at all. I guess you can always try a keylogger.
Anyway, all that aside. Right now this is not about being a people-pleaser and not rocking the boat. Right now, this is about YOU and what it will take for you to think the M is even worth trying to save. It is about YOU setting boundaries about what YOU WANT AND NEED.
"H, I was not clear enough. Either you participate with me in activities specifically meant to improve our M within the next month or we will not see our next anniversary," (again to steal the last bit from GEL). I believe Retrouvaille is a good place to start to try to enhance the real emotional and physical intimacy in our R. Thus, unless you come up with a viable alternative, that is what we need to get signed up for ASAP and attend together as a couple. I hope this is clear. It is non-negotiable, although I am open to alternatives to Retrouvaille that include a third=party working with us to improve our M."
P.S. I didn't see that your IC thought that H was involved in extracurriculars. I'd say ignore my wishy-washiness on what H is up to and go with your gut and IC.