Structural instability of critical quasicircle maps

Let $f:\mathcal{H}_q\to\mathcal{H}_q$ be a critical quasicircle map with irrational rotation number, and let $\mathcal{N}$ be the Banach neighborhood of $f$ consisting of nearby unicritical holomorphic maps. Let $\mathcal{S}\subset\mathcal{N}$ be the local quasiconformal conjugacy class of $f$. Structural-instability conjecture. The local conjugacy class $\mathcal{S}$ is a codimension-one submanifold of $\mathcal{N}$. In particular, critical quasicircle maps are structurally unstable. This is known for periodic-type critical quasicircle maps close to the associated renormalization fixed point and for critical circle maps with arbitrary irrational rotation number; the general case remains open.