Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following:

$$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $$

The problem is depicted for a jointly Gaussian distribution in the following figure:

The grey area in the figure corresponds to LHS, whereas the yellow area corresponds to RHS. My view is that since the distribution has higher values in the grey area, the inequality should be true. My first attempt at proving this was to use Gauss's Inequality, but have no idea how to apply it in this scenario. Any help or hint is appreciated.

A side question is do we really need the unimodality condition on the distribution or all symmetrical distributions can satisfy the inequality?