So is the schema for these branded tlds supposed to change from our current name.tld/page/subpage.html to page.name/subpage.html? That strikes me as a huge waste of time setting up subdomains for what are effectively just pages.
As an example of how it can be handled, Monash has https://my.monash.edu.au/ which is already known by the name my.monash; I expect that they will shift it to https://my.monash/ in the fairly near future.
What it ends up boiling down to for Monash University is that you can drop the .edu, .edu.au or .edu.my part. Oh, and you get to cough up a decent amount of dough for the endeavour (for the TLD application, for TLD maintenance and for software systems and such).