@cmuir
@jerry_nguyen is correct.
There is however a user javascript that will do exactly what you want and it works with assignments and classic quizzes. I do not knkow about new quizzes because I do not use them.
There is a canvancement written by james called quiz wiz: https://community.canvaslms.com/t5/Higher-Ed-Canvas-Users/QuizWiz-Enhancements-to-SpeedGrader-and-Qu...
the above link has information and what not about quiz wiz. the link below will take you to the install script page for quiz wiz. You will have to install tamper monkey first
https://github.com/jamesjonesmath/canvancement/tree/master/quizzes/quizwiz
then for the quiz wiz script, install the qw-rubrics option.
With the script enabled, not only will the rubric stay open, but there is an arrow that will perform the submission and move to the next student keeping the rubric open. It is quite the time saver.