The sensitivity-weighted variable bound for Boolean functions

Let $f:\{0,1\}^n\to\{0,1\}$ be a Boolean function. Write $n(f)$ for its number of relevant variables, $s(f)$ for its sensitivity, and define

$$S(f):=\sum_{i\in[n]}\frac{\delta_i(f)}{2^{\operatorname{sens}_i(f)}}.$$

Here $\delta_i(f)$ indicates whether variable $i$ is relevant and $\operatorname{sens}_i(f)$ is its individual sensitivity.

Sensitivity-weighted variable conjecture. For every Boolean function $f$,

$$n(f)\lesssim 4^{s(f)}.$$

More strongly,

$$S(f)\lesssim 1.$$

This would improve Simon's theorem by giving a constant bound on the sensitivity-weighted quantity and, consequently, the stated bound on the number of relevant variables. The paper presents this as an open conjecture and gives evidence for it, but no resolution is supplied.