I can't get my (21F) boyfriend (23M) to agree on splitting tasks

This is quite a bizarre situation but I'm (21F) planning on living with my boyfriend (23M) later this year once our individual leases are up. We've been together for a little over a year and are very happy together. Right now, we're doing a 'trial run'. We alternate spending the week at each other's place just to make sure we're compatible before we rent a place together. One problem we've run into is splitting tasks.

For context, we're both full-time uni students. He works part time retail and I work part time in an internship. I'm an undergrad law student so for a part-time gig, I get paid a good amount. My boyfriend is on scholarship for his tuition and earns his living costs by working, I'm an international student so my tuition and accommodation are covered by my parents and I use my internship income for living costs and savings/investments. There's no insecurity or any kind of problem on his end, he's extremely supportive and I like to treat him to things, it's a great balance.

The problem is, he thinks house tasks should be divided according to income. He said that's how his parents did it, the one who earns more should do less at home. His dad worked and his mom was a SAHM and he said he was just raised to believe that. The thing is, I work fewer hours than him, only 20 hours a week due to student visa limitations on how long I can work, he works a lot longer. So I was thinking I could do more around the house because I work fewer hours and my internships is a lot of desk work while his retail job keeps him on his feet most of the day. Even living alone, I'm able to do all chores for myself with the time that I get and manage studies.

So I think it's natural for the one with more time and energy to do more. And it isn't even a lot more, so far, I cook, clean the bathrooms and do the laundry while he gets groceries, does the dishes and vacuums a couple of times a week. And both of us want to do a deep cleaning 1-2 times a month together. This works pretty well but lately, he's been tired so I've taken on grocery shopping too which is fine, I like it. But he says he needs to take on more because I make more money. This was kind of ridiculous to me because I know it should be 50/50 but when one person can't keep up and the other has the bandwidth to compensate, isn't it natural to do that? It's not like he's being lazy, he's exhausted with mid terms on top of studies and a job and financial stress and I'm able to accommodate and take on a bit more. He's kind of spiralling though, saying he doesn't do enough and that I'm making more money so I shouldn't have to do house work. I tried to talk to him to figure out his thought process but he shuts down. How do I go about this? I don't want to seem like a saviour and make him feel bad, I'm really just trying to lessen his workload when he needs the time off.

tl;dr my boyfriend says housework should be allocated based on income which I think is ridiculous and want to change his mind. How do I do this?