Difference-product and difference-quotient sum-product conjectures

Let $A\subset\mathbb{R}$ be finite, and define

$$A-A=\{a-b:a,b\in A\},\qquad AA^{-1}=\{ab^{-1}:a,b\in A\},$$

with the convention $0^{-1}=0$. Also let $AA=\{ab:a,b\in A\}$. Difference-product and difference-quotient conjecture. For every fixed $\delta\leq 1$, one has both

$$|A-A|+|AA^{-1}|\gtrsim |A|^{1+\delta},$$

and

$$|A-A|+|AA|\gtrsim |A|^{1+\delta}.$$

These are sum-product-type expansion claims involving differences together with quotients or products. The candidate is presented without a name or resolution evidence, so its status remains open.