Well CX do if you don't have any status with them.
But remember, if your ticket is a paper one then the airlines can most likely only see the details if it is on their ticket stock, so you can give them the old ticket number even though that ticket doesn't actually have a coupon for the segment that you hold the reservation for... I have never had a problem doing this. In fact a tame travel agent can issue you a dummy ticket number to hold the reservation until you have a real ticket to enter there.