The last id engine released under the GPL was id tech 4, i.e. the Doom 3 engine (and the updated BFG version)*. id tech 5, 6, and 7 have since been used in Rage (2011), Doom (2016), and Doom Eternal (2020). Carmack was the only guy who both cared about open-source and had the juice to make it happen, but he left in 2013. So I kinda doubt it.