Here's a transaction made using the set up discussed in that thread: https://mempool.space/tx/1c41724a7b16ecd5e11867864d834eb24e9d22b372c86aa7869c4cc0b6b36d52. It signs from one mandatory key, and then also signs from a 2-of-3 multi-sig, essentially making it a 3-of-4 with one required key.

That's probably cheaper in size than the one proposed by hosseinimr93, but quite easy to mess with it during the beginning.

Just use taproot, and then your 4-of-6 transactions will be no bigger.