Degenerate stability of some Sobolev inequalities

We show that on $\mathbb S^1(1/\sqrt{d-2})\times\mathbb S^{d-1}(1)$ the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on $\mathbb S^d$. Our proof proceeds by an iterated Bianchi-Egnell strategy.