Am I in the wrong?

So I am genuinely asking for advice. When my boyfriend and I first got into our relationship we agreed that chores and bills will be split equally. We also agreed that if some days are harder for the other to be able to contribute half, that the other will help and pick up the slack.

We have been arguing for the past 3 days (on and off) about chores. We agreed that I will cook and he will wash dishes. Lately he has been complaining and down right not doing the dishes because I do not "set him up". He wants the kitchen to be clean, sink to be clean, and dishes rinsed or if needed to be soaked and stacked in the sink before he can start washing the dishes. I have always cooked, washed the dishes, and then cleaned and wiped down the kitchen. I was always raised that whoever did the dishes would also wipe down the sink after the dishes were washed and if dishes needed to be soaked it was the responsibility of the person doing the dishes.

I compromised and changed our chore list. I will cook and wash dishes every day and night, wash the clothes, and wipe the bathroom down throughout the week. He would mop once a week, deep clean the bathroom once a week, and fold clothes. Because he doesn't like the fact that some of my clothing needs to be hung he is now stating that it would be my responsibility to fold my own clothing. At this point I don't know how else to split the chores or how to make this equal. He states that he shouldn't have to do much since he has a harder job (I work in insurance claims and he has his own painting company.) Since it seems that he wants me to do more of the chores do you think it'd be ok for me to ask him to cover more of the bills? Or what other advice do you guys have?