Rule-based deletions from string list

I have a list of strings:

lis = {"a","b","c","12","d","q","r","X","s"}

I'd like to delete list members starting with "X" moving backwards through the list from "X" until a list member that's a digit character is found, to get:

res = {"a","b","c","12","s"}

Not sure how to use DeleteCases here?