Patterson–Wiedemann conjecture on the minimum Fourier amplitude of Boolean functions

Let $V_m$ be the $m$-dimensional vector space over $2$, and let $f$ range over all functions from $V_m$ to $\{\pm1\}$. Write $\widehat f$ for the Fourier transform of $f$ and $\|\widehat f\|_\infty$ for its supremum norm.

Patterson–Wiedemann conjecture.

$$\lim_{m\to\infty}\inf_f\frac{\|\widehat f\|_\infty}{2^{m/2}}=1.$$

This is the reformulation of the conjecture of Patterson and Wiedemann concerning the covering radius of the first-order Reed–Muller code, or equivalently the non-linearity of Boolean functions. The supplied text gives no resolution status.