If it's a typical single action rod it will not push on the skunk stripe and make it worse, and the rod should not get loose as long as there's some tension on it.
When tightened, the rod pushes toward the fingerboard and away from the stripe, so the popping out must be due to the wood drying out and shrinking or warping back into whatever shape it might have been before it was a guitar part.
Gibson rods and Fender cap fingerboard installed rods with no skunk stripe will push the filler strip out of place and get rattly if the filler strip (internal "skunk stripe") glue joint fails.
It would be good to get some glue in the joint to fix the loose skunk stripe, but that may be a job for a good tech, maybe the stripe needs to come out first to clean off old glue, and maybe it's just too small for the rout.
Not an expensive neck and still working though...