Can a sequence of conformal diffeomorphisms converge to something which is neither a diffeomorphism nor a point? This is a follow-up of this question.
Let $M$ be a compact oriented Riemannian manifold, of dimension $d\ge2$. Let $f_n \in \text{Diff}^+(M)$ be a sequence of orientation-preserving conformal diffeomorphisms which weakly converges in the Sobolev space $W^{1,p}(M;M)$ to a limit $f$. (for some $p>1$. I am ready to assume $p > \dim M$).

Is it true that $f$ must be either a constant map or a diffeomorphism?

(Or perhaps $f$ must be either constant or an immersion?). 
As an element in a Sobolev space, $f$ is only defined up to a measure zero set. So, the question is whether or not there always exists a representative which is smooth and is either constant or a diffeomorphism.
In this answer, a sequence of diffeomorphisms $f_n:\mathbb{S}^1 \to \mathbb{S}^1$ is constructed; the limit of $f_n$ maps half of the circle onto $\mathbb{S}^1$, and the other half to a point. I tried to build from this sequence a higher-dimensional example, but couldn't do so in a conformal way. (e.g. $f_n \oplus f_n$ or $f_n \oplus \text{Id}$ are not conformal).
 A: Let me convert my comment into an answer. 
Theorem. Let $C(M)$ be the conformal group of a Riemannian manifold $M$ with $dim(M)=n\ge 2$. If $M$ is not conformally equivalent to $S^n$ or $E^n$, then $C(M)$ is inessential, i.e. can be reduced to a group of isometries by a conformal change of metric.
This theorem has a long and complicated history (in particular, a long history of incorrect proofs), you can find its proof and historic discussion in
J. Ferrand, The action of conformal transformations on a Riemannian manifold. Math. Ann. 304 (1996), no. 2, 277–291.
Now, given this theorem, if $M$ is compact, unless $M$ is conformal to $S^n$, any sequence of elements $f_i\in C(M)$ subconverges to a conformal diffeomorphism. This leaves out the case of a conformal metric on $S^n$. The elements of $C(M)$, in this case, are (smooth) quasiconformal transformations (in the sense of the standard round metric on $S^n$). As such, they satisfy the "convergence property", i.e. every sequence subconverges either to a constant (uniformly on compacts away from one point) or to a quasiconformal  transformation. In the latter case, it is not hard to see that the limit is a diffeomorphism. In the former case, in order to get convergence in $W^{1,p}$ (most likely, $p=n$), one needs to work more and I do not have time for this. Check out books on quasiconformal maps, for instance
Iwaniec and Martin, "Geometric Function Theory and Non-linear Analysis", Oxford University Press, 2002.
Also, note that every sequence of qausiconformal mappings of $S^n$ converging to a constant can be written as a composition $f_n\circ h_n$ where the sequence $h_n$ converges to a quasiconformal mapping and $f_n$ is a sequence of Moebius transformations which converges to a constant map (away from one point). This effectively reduces the problem to the case of mappings of the form $x\mapsto \lambda_i x$, $x\in R^n$, $\lambda_i>0$ is a sequence diverging to infinity. (One also has to check how $W^{1,p}$-convergence behaves under the composition.)  
