On all but one of the sites we've updated to 8.1.0.b1 running the Core package, the "Migration necessary" notification will not go away, no matter how many times we click the "More Details" button, or the "Start Migration" button on the migration page. The one site running the Pro package (not on this same server, FWIW) doesn't persistently show it.
For the most part, the sites are all on a combination of PHP 7.4 (because PHP 8 compatibility in the WordPress ecosystem is still a joke) and WordPress 6.3. There are a couple of outliers here; one site's on WP 5.5 and two sites are surprisingly running on PHP 8.0 (none of the outliers are the site where the notification cleared itself).