Lev and Sonn's difference-set conjecture for finite-field squares

Let $p$ be a prime, and let $\mathcal{R}_p$ denote the set of nonzero squares in ${\mathbb F}_p$. Lev and Sonn's conjecture. For all sufficiently large primes $p$, there is no subset $A\subseteq {\mathbb F}_p$ such that

$$A-A=\mathcal{R}_p\cup\{0\}.$$

Kalmynin resolved this difference-set conjecture and proved a generalization for proper multiplicative subgroups of prime fields, so the claim is no longer open.