The entropy–KL-divergence strengthening of the union-closed sets conjecture

Let $A$ and $B$ be independent identically distributed samples from a distribution supported on a family of subsets of $[n]$. Write $H(A)$ for the entropy of $A$, and $D(A\cup B\mathbin{\|}A)$ for the Kullback–Leibler divergence between the distributions of $A\cup B$ and $A$. Entropy–KL strengthening. If

$$\Pr[i\in A]<0.5$$

for every $i\in[n]$ and $H(A)>0$, then

$$H(A\cup B)+D(A\cup B\mathbin{\|}A)>H(A).$$

The paper presents this as a sufficient strengthening that would imply the union-closed sets conjecture; it is not resolved by the paper and is therefore open.