No, it absolutely wouldn't happen that way. Nothing you do to one end of the rod can affect anything at distance L in time les than L/c, where c is the speed of light.
Does that mean the rod can't be rigid, no matter what it's made of? Yes. The forces between the parts of the rod are all fields that cannot travel faster than light. So the rod can't be rigid.
Mike W.
(published on 04/26/2019)