Very unlikely.
If they do, then it will open the door for the defense to introduce all of the evidence about what a good boy Trayvon was not, such as the fight videos, the school suspension, the marijuana use, etc.
Your "etc" would include a backpack containing stolen jewelry and a screwdriver, wouldn't it? Zimmerman thought the kid was casing the neighborhood, looking for a good place to burgle, which is exactly what a would-be burglar does, hoping to find signs of nobody home.