Conjectured improved Junta approximation for low-influence Boolean functions

Let $f\colon\left\{0,1\right\}^{n}\to\left\{0,1\right\}$ be a Boolean function, write $\mu=\mathbb{E}[f]$, and let $I(f)$ denote its total influence. A function $g$ is a Junta if it depends on only a bounded number of coordinates. Improved Junta approximation conjecture. For any $M,\epsilon>0$, any function $f$ satisfying

$$I(f)\le 2\mu(\log(1/\mu)+M)$$

can be $\epsilon\mu$-approximated by a function $g$ that depends on at most $O(\log(1/\mu)\cdot 2^{M/\epsilon})$ coordinates. The paper presents this as a reasonable conjecture in its open-problems section; it would strengthen the paper's Junta approximation result for functions whose influence is close to the minimum.