The entropic Solymosi inequality

Let $X$ and $X'$ be independent and identically distributed, discrete, real-valued random variables with finite entropy $H(X)<\infty$. Entropic Solymosi inequality. One has

$$2H(X+X')+H(XX')\geq (4-o(1))H(X),$$

where $o(1)$ tends to $0$ as $H(X)\to\infty$. This is the entropy analogue of Solymosi's combinatorial sum-product inequality and would imply the entropic sum-product conjecture with $\delta=\frac13$. The source states that this inequality is not proved; it is expected to be sharp in view of the known bound $\delta\leq\frac13$.