Iosevich's sum-product conjecture over prime fields

Let $A\subset\mathbb{F}_p$, and let $\epsilon>0$. Iosevich's conjecture. If

$$|A|\gg p^{\frac{1}{2}+\epsilon},$$

then

$$A\cdot A+A\cdot A=\mathbb{F}_p,\qquad (A-A)^2+(A-A)^2=\mathbb{F}_p.$$

The conjecture concerns simultaneous additive-multiplicative expansion in prime fields and is stated after a theorem giving a weaker exponent for related product-set conclusions; the source does not state a resolution status.