The problem is not that he is a liar. The problem is what is causing him to lie. That is what you need to deal with. As the CL on the ADHD forum. The first thing that comes to mind is its a way to cope with ADHD or ADD. The second thing is dealing with his home situation.
How are his grades? Not being able to answer simple things like closing his door are pretty typical ADHD kind of things. Does he have trouble doing homework? Does he "forget" to do things you have asked him to do? Do his teachers complain about him not paying attention? Oh, has he been consistently at the same elementary school?
I can give you a lot more info on this whole ADHD or ADD thing if any of the above questions ring a bell.
I made an accound to answer your question while reading through these. I'm 37 Years old and from the age of 9 to sometime in my 20's I had two best friends. The one I will discuss we will call Clyde the other Ramond. Clyde is who I will focus on. Clyde sounded a lot like your stepson. From the time I knew Clyde he would lie about all kinds of stuff. He would embellish and make up stuff and stories that were not true. For instance we came back from the mall and were in the kitchen talking to Clydes parents we were probably about 12 13 at the time this would have been 1993 1994 Planet Hollywood had just opened at the Mall of America which was owned by Stallone, Willis, and someone else. It was a normal conversation and for no reason he just lied and told his mom that we saw Bruce Willis at the mall and there was a huge crowd of people chasing him through the whole mall it was insane. I knew this wasn't true cause I was there but he would do **** like that, make up totally, or embellish stories and then I'm there, either needing to Call BS on it or go alone with his lie. I got dragged into for no reason. Clyde was a good looking guy, fun to be around, no problem getting girls, but seemed to lack the ability to really connect with any significant other throughout his life, he was married then separated or divorced and now maybe back with his wife. He actually started doing drugs pot, coke whatnot around 7th or 8th grade and ran from the police he was a bad kid. He had a younger sister and two older brothers both parents not divorced and still happily married and a good life good upbringing and good family none of his other siblings were bad at all. He was smart and intelligent. He ended up going to a youth jail type place cause his parents couldn't get a hold on him for a year or 2 around age 14 or to age 16 or so. But the first time I chewed tobacco was with Clyde in 5th grade at the public swimming pool it was his idea he got me to do it. He was a bad influence, he was also the kid who would do the crazy **** in the group and was just more out of line then the rest of the kids we hung out with. The first time i egged a house was at the age of 12 or so at a sleepover at his house. Basically what I'm saying is I checked a list of items to check if your a sociopath and you could check every box for Clyde. IDK if that helps you at all but thats my story