Almost commuting unitaries with spectral gap are near commuting unitaries

Let $\mathcal{M}_n$ be the collection of $n\times n$ complex matrices equipped with operator norm. Suppose $U, V \in \mathcal{M}_n$ are two unitary matrices, each possessing a gap larger than $\Delta$ in their spectrum, which satisfy $\Vert UV-VU\Vert \le \epsilon$. Then it is shown that there are two unitary operators $X$ and $Y$ satisfying $XY-YX = 0$ and $\Vert U-X\Vert + \Vert V-Y\Vert \le E(\Delta^2/\epsilon) \left(\frac{\epsilon}{\Delta^2}\right)^{\frac16}$, where $E(x)$ is a function growing slower than $x^{\frac{1}{k}}$ for any positive integer $k$.