The half-density conjecture for sum-free subsets of symmetric convex regions

For a convex region $R\subset\mathbb{R}^d$, let $\sigma(R)$ denote the maximal proportion of the volume of $R$ occupied by a sum-free subset of $R$. A region is symmetric if $R=-R$.

Half-density conjecture. For every symmetric, convex region $R\subset\mathbb{R}^d$,

$$\sigma(R)\leqslant \frac{1}{2}.$$

The bound is known for the cubes $[-1,1]^d$ by the theorem proved in the paper, while the conjecture concerns arbitrary symmetric convex regions.