I can't find anything on differences in indoor hygiene causing a difference to the immune system. Homeschooled children will still be exposed to the same viruses through their parents or anyone else. It may very well be a misunderstood study from the 80s. It is currently neither confirmed or proven wrong.

However I've found a slightly different study. It compares children who go to conventional kindergartens to children in outdoor kindergartens. I'm not sure if you're familiar with that these "forest-kindergardens", but the general idea is that the kids are outside all day. Due to the need for warm clothing and constantly getting dirty, they usually wear the same outdoor suit throughout the week. They're exposed to natural bacteria a lot more, but less so to airborne viruses.

So it's not exactly the same thing you asked for, but it does show a difference in their immune systems.

https://www.science.org/doi/10.1126/sciadv.aba2578

Another study showing the same thing through a different comparison of children in rural vs. suburban areas.

https://www.ucc.ie/en/news/2023/rural-environment-supports-childrens-immune-systems.html