Like @Coop, I went back and looked at the original pic as well. Because of the paint, it's difficult to tell if it's MDF but, if it's a stock cabinet, chances are it is MDF. It also looks like you have room to install about a 2" X 2" block on the sides. You'd have to measure to be sure. Then, just a couple screw up through those in some oversized holes. Or, you could use the figure 8's in the new blocks. 3/4 is enough for figure 8's but, I'd go thicker if you have the room.
Silicone would also hold and wouldn't restrict movement. I'm just not sure how well it would hold up.