Can we prove the following claim?
For any real function $f(x)>0$ and any complex functions $\phi_1(\mathbf{x})$ , $\phi_2(\mathbf{x})$ such that the following integral has finite result, then $$\int d^3 x d^3 yf(|\mathbf{x}-\mathbf{y}|) \phi_1(\mathbf{x})\phi_2^*(\mathbf{x})\phi_2(\mathbf{y})\phi_1^*(\mathbf{y})>0.$$ The range of integral is $\mathbb{R}^3$.
If it's true, how to prove it. If false, give me an counterexample.
Expanding all functions in terms of their Fourier transforms $F(x) = \int \hat{F}(k) e^{-ikx}{\rm d}k$ and evaluating the integrals over $x$ and $y$ (which gives us $\delta$-functions) we obtain
$$\int f(x-y)g(x)g^*(y)\,{\rm d}x\,{\rm d}y = \frac{1}{(2\pi)^3}\int |\hat{g}(k)|^2\hat{f}(-k)\,{\rm d}k~~~~\text{where}~~~g(x) = \phi_1(x)\phi_2^*(x)$$
which is guaranteed to be positive as long as the fourier transform of $f$ is positive $\hat{f}\geq 0$. For example for the special case $f(x) = \frac{1}{|x|}$, which is the case that most often occurs in practice (chemistry / physics), this is the case as $\hat{f}(k) = \frac{4\pi}{|k|^2}$.
A sufficient condition for the Fourier transform of a radial function $f = f(|x|)$ to be positive is that $|x|f(x)$ is decreasing. See for example E.O. Tuck "On Positivity of Fourier Transforms" and this MO question.