Qualitative stability for a family of trace Sobolev inequalities

The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen & Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent paper of Fan, Li & Zhang, and in conjunction with their local analysis, yields sharp quantitative stability for this family of inequalities when $p=2$.