How long should I roll a die?

I roll a die. I can roll it as many times as I like. I'll receive a prize proportional to my average roll when I stop. When should I stop? Experiments indicate it is when my average is more than approximately 3.8. Any ideas?

EDIT 1. This seemingly easy problem is from "A Collection of Dice Problems" by Matthew M. Conroy. Chapter 4 Problems for the Future. Problem 1. Page 113.
Reference: https://www.madandmoonly.com/doctormatt/mathematics/dice1.pdf
Please take a look, the collection includes many wonderful problems, and some are indeed difficult.

EDIT 2: Thanks for the overwhelming interest in this problem. There is a majority that the average is more than 3.5. Some answers are specific (after running programs) and indicate an average of more than 3.5. I will monitor if Mr Conroy updates his paper and publishes a solution (if there is one).

EDIT 3: Among several interesting comments related to this problem, I would like to mention the Chow-Robbins Problem and other "optimal stopping" problems, a very interesting topic.

EDIT 4. A frequent suggestion among the comments is to stop if you get a 6 on the first roll. This is to simplify the problem a lot. One does not know whether one gets a 1, 2, 3, 4, 5, or 6 on the first roll. So, the solution to this problem is to account for all possibilities and find the best place to stop.