I was a big fan of our previous dining room rug, but unfortunately the cats were as well and chose to mark their territory accordingly. Despite numerous attempts to purge the smell, it never really worked, and so I opted to remove it. Once I did that, the smell immediately went away, so thankfully nothing had seeped into the floorboards.
I decided to get a replacement rug from Ruggable, since it would be washable if the cats decided to attack again. While their styling options are a little bit more limited, I still think the end result looks good and should be significantly more durable.