Sharpness of rigidity estimates for pairs of quasiregular mappings

Let $B_1$ be the unit ball in $\mathbb{R}^2$, let $Q$ be fixed, and let $S(A)$ denote the symmetric part of a matrix $A$. A pair of maps is $Q$-quasiregular as in the paper, and $SO(2)$ is the group of planar rotations.

Sharpness conjecture. There exists a sequence of positive numbers $\epsilon_k\rightarrow 0$ and a sequence of pairs of $Q$-quasiregular maps $u_k:B_1\rightarrow \mathbb{R}^2$, $v_k:B_1\rightarrow \mathbb{R}^2$ with

$$\int_{B_1} |Du_k|^2\,dz\leq 1$$

such that

$$\int_{B_1} |S(Du_k)-S(Dv_k)|^2\,dz=\epsilon_k$$

and

$$\int_{B_{\frac{1}{2}}} |Du_k-R_{\theta}Dv_k|\,dz\geq 1\quad\text{for all }R_{\theta}\in SO(2).$$

This asserts that the rigidity conclusion cannot hold with an error tending to zero under these hypotheses, and is intended to show that the estimates proved in the paper are essentially sharp. The parser provides no evidence that this statement has been resolved.