The power difference set conjecture for multiplicative subgroups of finite fields

Let $p$ be prime, and let $K$ be the multiplicative subgroup of $\ell$-th powers in $\mathbb{F}_p$. A power difference set conjecture asserts that if $K$ forms a difference set of $\mathbb{F}_p$, then

$$\ell=2,4,8.$$

The power difference set problem asks when the subgroup of $\ell$-th powers is a difference set; it is known that $\ell$ must be even, but the stated classification remains open.