Don’t bite Oda’s bait with Carrot. There’s a reason he had Neko bring up her relationship with the Straw Hats specifically and had Carrot looking uncertain in this panel. It’s all done purposefully.
This is his way of telling you that Carrot will do the complete opposite of what they want her to do.
Oda has never had a situation where a character did precisely what OTHERS wanted them to do and not what THEY wanted to do without it leading to some conflict whether minor or major. Vivi remained behind not because Cobra and his assistants wanted her to do, but because Vivi HERSELF wanted to. Hell, look no farther than Luffy. Garp prepared the way for Luffy to become a marine, but Luffy went in the opposite direction to pursue HIS dreams, not Garp’s, even IF the latter’s intentions were well meaning.
Don’t. Be. Fooled.