You are just being unreasonable. Don't you know housework is really hard work. We are SURVIVORS damn it!
Sorry I just couldn't help myself.
Has he ever ever helped before? Did he ever have to do housework before you got involved with him? Maybe the concept is foreign to him.
I'm not great with housework but since I live alone there is only one person to do it. Usually I put it off if I can. No one but me to complain though. But I do have a full time job. In the winter I do an hour on the treadmill after work. From spring until fall I bike to work (10 km) then spend 9.5 hours at work thn bike home again. Chronic back pain and poor sleep (5.5 hrs/ night on average) leaves me fatigued. So I don't really feel like doing it. But I still cook my own meals and clean up after myself. And that just reminded me that I forgot to bake muffins. I freeze them and have one at work for my first break.
I know this won't go over well but I'll say it anyway. You and H should fill out time sheets. Record what you do all day. Then have a review at the end of the week.
Maybe you have to get rid of the TV.