There is no such world. If such world existed you think oda would just make carrot look bad for no reason?
Now that is an interesting point ! In a world where Carrot was portray with the strawhat for an entire arc fighting and playing with them, a full alliance member, why do you think that she is part of the very few people that got defeated in the story of Onigashima ?
Strenght ? Powerlevel ? I sence something deeper here.. like a good reason.. why would Carrot become Queen on a defeat ? Isn't that odd ?
I sence something strange here but just a feeling hehe..