By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty$ (in the same homotopy class as $f$) whose Beltrami coefficient $\mu$ has the smallest $L^{\infty}$ norm $x$.

My question is : If $g_n$ are q.c homeomorphisms fixing $0$, $1$, and $\infty$ in the same homotopy class as $f$ such that the $L^{\infty}$ norms $x_n$ of their Beltrami coefficients $\mu_n$ converge to $x$ (but we don't know whether the $\mu_n$ converge to anything), then do the $g_n$ converge to $g$ in some space (ex: Sobolev space)?

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty$ (in the same homotopy class as $f$) whose Beltrami coefficient $\mu$ has the smallest $L^{\infty}$ norm $x$.

My question is : If $g_n$ are q.c homeomorphisms fixing $0$, $1$, and $\infty$ in the same homotopy class as $f$ such that the $L^{\infty}$ norms $x_n$ of their Beltrami coefficients $\mu_n$ decrease to a limit $x$ (but we don't know whether the $\mu_n$ converge to anything), then do the $g_n$ converge to $g$ in some space (ex: Sobolev space)?