Interesting. If I lock with the remote and the top down, the glove box locks. If I unlock the remote once (driver's side door) the glove box unlocks. But I hadn't thought to check this with the center main lock button. Both of my 2001's had broken wires inside the glove box latch which I had to re-solder. (And there's something about locking the glove box with the handle half open - which seems to put my car in valet mode. But I've not quite figured this out yet.
I'll get back to you on this.