The independence conjecture for sum-free inverse-closed subsets of finite fields

Let $\mathbb{F}$ be a finite field. Write $\sigma(\mathbb{F})$ for the density of the largest sum-free subset of $\mathbb{F}$, and $\mu(\mathbb{F})$ for the density of the largest subset of $\mathbb{F}$ that is both sum-free and closed under multiplicative inverses.

Independence conjecture.

$$\mu(\mathbb{F})=\sigma(\mathbb{F})^{2}+o(1)$$

as $|\mathbb{F}|\mathbin{\to}\infty$.

The conjecture expresses the expectation that the properties of being sum-free and having a sum-free inverse set behave independently. It would make the previously established lower bounds in characteristics $2$ and greater than $2$ asymptotically sharp.