Yes. A proof is recorded in Prop 1.4.6 and Rmk 1.4.7 of my thesis, though as Daniel's answer indicates, the experts were aware of this much earlier. It also appears that I upvoted at some point, so I think I was aware of Daniel's answer here when I wrote my thesis.

Yes. A proof is recorded in Prop 1.4.6 and Rmk 1.4.7 of my thesis Free duals and a new universal property for stable equivariant homotopy theory, though as Daniel's answer indicates, the experts were aware of this much earlier. It also appears that I upvoted at some point, so I think I was aware of Daniel's answer here when I wrote my thesis.