This idea came to us via my brother Jason during Christmas dinner, and is a great one. Basically make the cardboard center roll for toilet paper out of some steam-resistant water-soluble material that would dissolve near instantly in water. When you are done with the last piece of paper, you just throw the roll tube in the toilet saving landfill and garbage space, or composting space if your town supports it.
Or, expanding on the idea, why not eliminate the cardboard center tube entirely? If you make the center of the roll just slightly bigger than the standard holder size (about 3/4 inch), the structure of the roll would likely stay intact until very near the last sheet, which is always ruined by the glue that holds it to the cardboard anyway. The added benefit is that you can fit more toilet paper per roll, reducing shipping costs. If the manufacturing requires a structure to be in place to support the center of the roll, some small compostable grommet could be used, and would be popped out by the roll holder when the roll is installed.
Addendum: Apparently the first part of this one has been tried, making the second option moot:
I disagree with the article’s point, and the third option above (no roll center tube) should still be kept on the table.