The Gopalan-Servedio conjecture: is $\sum_i\widehat f(\{i\})=O(\sqrt{\deg f})$ for Boolean functions?

Theorem 1: for every $C>0$ there is a nonconstant $f:\{-1,1\}^n\to\{-1,1\}$ with $\sum_i\widehat f(\{i\})>C\sqrt{\deg f}$; equivalently $\sum_i|\widehat f(\{i\})|/\sqrt{\deg f}$ is unbounded over Boolean functions of positive degree, so no constant multiple of the conjectured bound holds. The dimension and copy counts in the construction are finite but not usefully bounded, so no growth rate of the violation in terms of degree is given. The true order between $\sqrt{\deg f}$ and the linear bound $\deg f$ is left open.

Theorem 1: for every $C>0$ there is a nonconstant $f:\{-1,1\}^n\to\{-1,1\}$ with $\sum_i\widehat f(\{i\})>C\sqrt{\deg f}$; equivalently $\sum_i|\widehat f(\{i\})|/\sqrt{\deg f}$ is unbounded over Boolean functions of positive degree, so no constant multiple of the conjectured bound holds. The dimension and copy counts in the construction are finite but not usefully bounded, so no growth rate of the violation in terms of degree is given. The true order between $\sqrt{\deg f}$ and the linear bound $\deg f$ is left open.