After about a month of this not getting any better I decided to force the issue. I told him he would have to go in the playroom for 5 minutes every day and play by himself. The first day he stood in the doorway the entire time. He did this for a few days. Then one day I told him he wasn't coming out if he hadn't played at all, even if the timer went off. So he played. And he played the next day. After that I increased the time each day until we were up to an hour. Sometimes he even fell asleep. He started to not like it if we were busy running errands and didn't have time for Quiet Time.
I started the whole process for very selfish reasons. I needed some respite from him during the day. But in the end, he benefitted from it so much more than I did! He developed an imagination. He started playing in other areas of the house. He was less fearful about being alone. I saw such changes in him, I knew I would use this technique with any more children that came into my home.
So now that Mia doesn't always take a nap, we have graduated to Quiet Time. After lunch she gets to load up a rattan tray with play items and at least one book (my requirement). I then set the timer for one hour. She plays, sings, and sometimes falls asleep. I have noticed that her imaginative play has increased at other times of the day as well.
Love love love