Uniqueness of diffeomorphic minimizers of $L^p$-mean distortion

We study the $L^p$-mean distortion functionals, $${\mathcal{E}}_p[f]={\int }_{\mathbb{Y}}{K}_f^p(z)dz,$$ for Sobolev homeomorphisms $f:\bar{\mathbb{Y}}\stackrel{\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}}{\to }\bar{\mathbb{X}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0\text{\hspace{0.17em}⁣}:\mathrm{\partial }\mathbb{Y}\to \mathrm{\partial }\mathbb{X}$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1<p<\mathrm{\infty }$, the functional ${\mathcal{E}}_p$ admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique.