Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates
Gil, would something like this be a counterexample?

Let $m$ be such that $n=m+\log m$, and think of an $n$-bit input as being a pair $(x,i)$ where $x$ is an m-bit string $(x_1,\ldots ,x_m)$ and $i$ is an integer in the range $1,\ldots,m$ written in binary.

Then we define $f(x,i):= x_1 \oplus \cdots \oplus x_i$

Now for each $i=1,\ldots,m$ the function f() has $1/m$ correlation with the Fourier character $x_1 \oplus \cdots \oplus x_i$, and so the "level i" has at least a $1/m^2$ fraction of the mass. (In fact more, but this should suffice)

f() can be realized in depth-3: put all the XORs in a layer, and then do the "selection" in two layers of ANDs, ORs and NOTs (not counting the NOTs as adding to the depth, as usual).