Burkholder-function integral conjecture for the Beurling–Ahlfors inequality

Let $U$ be the Burkholder function appearing in the source, let $\partial$ and $\overline{\partial}$ denote the complex derivatives, and let $f\in C_0^{\infty}(\mathbb{C})$. Burkholder-function integral conjecture.

$$\int_{\mathbb{C}}U(\overline{\partial}f,\partial f)\,dm(z)\leq 0.$$

The inequality is proposed because the pointwise comparison of the relevant Beurling–Ahlfors function with $U$ would make it imply the Iwaniec conjecture. The source gives no resolution.