What circumstances is treated lumber the preferred choice on ANY kind of project. I assume fence posts is NOT one of them. Never seen a straight fence post made out of treated lumber. On cedar fences here, theybare now all built with galvanized pipe posts.
I used treated lumber to build my 16'x28' deck, the deck boards are cedar and all of the upright posts were wrapped in 1/4" cedar cladding. The cedar was coated with Cutek oil. Basically, the PT lumber was used only for the framing.
PT treated lumber does not change the properties of the wood itself, the penetration is only so deep. The purpose is to protect it from the elements where untreated lumber would otherwise decay rapidly.
I use minimum 6"x6" for fence posts, a 4"x4" just doesn't cut it IMO.