Characterization of balanced three-variable integral kernels

Let $\alpha\in(0,1)$, and let $f:[0,1]^3\to\mathbb C$ be a symmetric integrable function such that

$$\int_{A\times A\times A^{\mathsf c}} f=0$$

for every measurable $A\subset[0,1]$ with $\lambda(A)=\alpha$.

Three-variable kernel conjecture. If $\alpha\notin\{1/3,2/3\}$, then $f=0$ almost everywhere. If $\alpha=1/3$, then, almost everywhere,

$$f(x_1,x_2,x_3)=g(x_1,x_2)+g(x_1,x_3)+g(x_2,x_3),$$

where $g$ is a symmetric function on $[0,1]^2$ satisfying $\int_0^1g(x,y)\,\mathrm{d}y=0$ for every $x$. If $\alpha=2/3$, then, almost everywhere,

$$f(x_1,x_2,x_3)=g(x_1)+g(x_2)+g(x_3),$$

where $g$ is a function on $[0,1]$ satisfying $\int_0^1g(x)\,\mathrm{d}x=0$.

The displayed forms are motivated by explicit counterexamples at $\alpha=1/3$ and $\alpha=2/3$. The source explicitly leaves this conjecture open, and notes that it would imply further quasi-randomness results away from the exceptional values.