Sarwate's conjecture on Fourier coefficients of power permutations

Let $L$ be a finite field of characteristic $2$ with $q=|L|$, and let $f(x)=x^s$ be a power permutation of $L$. Define its Fourier coefficients by

$$\widehat f(a)=\sum_{x\in L}\mu(ax+f(x)).$$

Sarwate's conjecture. If $[L:{\mathbb F}_2]$ is even, then

$$\sup_{a\in L}\widehat f(a)\geq 2\sqrt q.$$

The source identifies this as Sarwate's main global conjecture and explicitly says that it is still open; it follows discussion of results proving an earlier characteristic-two conjecture.