That looks like lungwort. The only reason I know that is because my wife has a little planted in the rock gardens. It has not spread or become a problem, so we've never considered spraying it. I have no Idea if Raptor, 24,DB, Brassigan or other clover-safe herbicide will kill it. But at least we've identified what it is. You can fine tune your herbicide search.
I know a lot of guys like to mix chicory in their plots, but it sure complicates herbicide choices. I seldom plant it any more.
And if I do have any growing, I'm not afraid to sacrifice it. My hate for CERTAIN weeds (not all weeds) far outweighs my love for chicory.
4 to 6 oz. per acre of Slay/ Pursuit/ Thunder should clean those weeds up and your chicory will probably survive with some burned edges. I use Thunder which I buy for $235 a gallon at Daniels Farm Store 717-656-6982