Low-influence approximation conjecture for Boolean functions

For a Boolean function $f$, define its total influence by

$$I(f)=\sum_S\widehat f(S)^2|S|.$$

A Boolean function $g$ $\epsilon$-approximates $f$ when $\mathbb E(f-g)^2\le\epsilon$. Low-influence approximation conjecture. For some absolute constant $C$, every Boolean function $f$ can be $0.01$-approximated by a circuit of depth $d$ and size $M$ satisfying

$$(\log M)^{Cd}\le I(f).$$

This is proposed as a reverse-Håstad-type statement and remains open in the supplied text.