I've tried running some in 3 deep with mixed results. In the years that I've tried it, it seemed like a boomer colony like this really needed a lot of overhead in the way of resources to support that big of a colony, and it seemed like my 2-deeps outproduced my 3-deeps. I know this goes against common logic, but it's what I saw when I tried it. 3-deeps are great for pulling splits from, and it's nice not having to go down into that bottom brood box. In my case, the 3rd deep gets backfilled during the summer if you don't rotate empty comb into it too.
Try it if you'd like and compare results with your 2-deeps. Results with this approach will vary depending on many factors including location, type of bees that you have, forage availability & weather, amount of time that you have to invest on management, temperament of bees, hive ventilation techniques, alignment of the planets, whether or not you paid your taxes on time, etc.
See how it works for you and let everyone here know what your results were.