  In this work we prove a sharp quantitative form of Liouville's theorem, which
asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$
with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this
estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch. Ration.
Mech. Anal. 239(1):219-299, 2021), with different proofs given later on by
Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq
4$ requires new arguments because it is genuinely nonlinear: the linearized
version of the estimate involves quantities which cannot control the distance
to M\"obius transformations in the conformally invariant Sobolev norm. Our main
tool to circumvent this difficulty is an inequality introduced by Figalli and
Zhang in their proof of a sharp stability estimate for the Sobolev inequality.
