I have an 11 year old and a 7 year old. I use to have the same problem with them. They would walk through the door and the "verbal bombs" would just start.. I mean as SOON as they walked through the door it was constant fighting and bickering and interrupting one another.
I sat them down after they started one day and told them that I am sick of them coming home, and as soon as they hit the door, the verbal bomb starts. She ran away from me, she hit me, she didn't do this, she did that, she...NO the verbal bombs stop...I don't care if she ran away from you, you both know to keep your hands to yourself, unless one of you does something that is putting one of you in mortal danger..I DON'T WANT TO HEAR IT! Work it out between the two of you, don't interrupt each other, and talk nicely. If she is safely walking down the sidewalk and is two steps in front of you...I DON'T WANT TO HEAR IT, if she decided to talk to her friend for 10 seconds instead of waiting for you...I DON'T WANT TO HEAR IT.
Now, if anyone starts verbal bombs when they walk through the door...they are sent to a room...both of them and they are to work out their differences and until they do, they are not to come out. I do not need to solve every little indifference they have with each other, they need to figure out how to do it on their own.
A chore chart/reward chart doesn't work, believe me I've tried them. They just reinforce the fact that they are going to get something in the end, and sometimes they reinforce the negative behavior. Think about it, your daughter does her chore and doesn't get money for the day she lied...but she still gets money in the end, so in her mind, who cares. No, it shouldn't work that way. If you are going to continue with the chore chart then if she lies or fights you about doing her chores then she should get nothing for the whole week.
Eventually you will get to the point where it is just not worth it and throw them away. When they are doing good for the week you can go out and get them some nail polish or give them some money, and tell them you are proud of them for doing everything so peacefully this week.