Linear Coefficients versus Total Degree

Prove that for every Boolean function f: {−1,1}^n → {−1,1}, the sum of its linear Fourier coefficients satisfies ∑_{i=1}^n ĥf(i) ≤ √(deg(f)).

Background

This inequality would relate the weight of degree-1 Fourier mass in the signed sum to the total degree of the Boolean function, strengthening the trivial bound via total influence and degree.

Majority on k variables suggests a potentially tighter form, but currently no bound better than the trivial TInf[f] ≤ deg(f) is known to control ∑ ĥf(i).