Positive-influence conjecture for simply-rooted Boolean functions

Let $\{0,1\}^n$ be the Hamming cube, and let $f:\{0,1\}^n\to\{-1,1\}$ be a simply-rooted function, as defined in the paper. Let $\widehat f(\emptyset)$ denote its level-zero Fourier coefficient, let $I^+(f)$ denote its positive influence, and let $k\in[0,n-1]$ satisfy

$$\widehat f(\emptyset)\leq -(1-2^{-k}).$$

Positive-influence conjecture. One has

$$I^+(f)\leq (k+1)2^{-k}.$$

The conjecture proposes a sharp upper bound on positive influence in terms of the negative level-zero Fourier coefficient. The examples $C_k$ described in the paper attain the bound, but the conjecture remains open; the corresponding edge-isoperimetric lower bound concerns total influence rather than positive influence.