Harlock: Nail polish remover has acetone, or most do. Acetone can possibly dissolve your mat.
I had this issue, and heres what I did...
Overlay the offensive lines, a segment at a time, with the SAME dry erase marker, and color. Immediately douse with alcohol, like with a cotton ball, and make it quite moist. Rub vigorously with the tip of your finger, or fingernail, if need be.
You may have to wash, rinse, and repeat. You should get the offensive lines to a barely noticeable level. I got mine to where you can only see the lines in a certain light.