(this may not always happen)

1. put a Saddle on pig(dont show him a carrot or a carrot on a stick before you ride him for the first time)

2. Ride on the pig

3. Now show him a carrot on a stick for the first time

Tthe pig wont move, it doesnt matter if you will be in gamemode 1, gamemode 0 or if the Carrot on a Stick is in the main hand or the second hand, he will just eat(not always) it and and wont move, the pig's head is not moving too. Its like the pig is statue. The pig might start moving after something like half a minute but it might not move at all and stay like this

When I dont ride the pig and holds the Carrot on a Stick, the pig is following me and looking at me.