Painting over magic marker

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I let my daughter use paint pens on her walls. Come to find out she also used permanant marker as well and I now have several spots that I can't cover. I've put 2 heavy coats of good primer on it but still several areas have leached through. I dread the thought of having to use an oil based primer. Is there any chance that the paint itself will cover the remaining stains? Any suggestions for the easiest way to cover this or am I doomed to the oil based primer? I'm planning on using antique white.