Unlikely, unless you lose a combat badly. You want to keep them under the eyes of the general and BSB because of stupidity; they're highly unlikely to panic under LD 10 rerollable.
Unless you lose combat. Against equivalent hard-hitting unit, say Dwarf Hammerers or Dragon Princes, they could lose enough to drop that Ld10 down to 5 or 6 and then you're in serious trouble.
Yes, that's what I meant when I said "... unless you lose a combat badly..." at the beginning of my post.
Given equal points, Dragon Princes would have to achieve a decisive advantage on the charge, though. If they don't, following rounds favor the COK significantly with 3 S4 attacks vs. 3 S3 from the Dragon Princes. The difference between a 2+ save and a 3+ save after STR mods is pretty hefty.
Of course, the presence of characters and magic complicates the matter...
You're forgetting that Dragon Princes can attack in 3 rows. So they'll be churning out more rider attacks. So I think that would be the deciding factor. On the charge, the higher WAS of Dragon Princes would Trump that of the Cold One attacks. And with a higher movement rate of the DP horses, they achieve a successful charge more easily, gaining the Lance bonus.
That being said, after the charge round, I think COK get better, but that might wash out as well due to the DPs 6++ ward save.
It's all REALLY well balanced if you think about it! Those truly ARE mirror image units!
My armies: Empire, Dwarfs, High Elves, Wood Elves, Dark Elves, Lizardmen, Skaven, Tomb Kings, Ogre Kingdoms, Orcs and Goblins, and Skitarii (40K).