Rivest's optimality conjecture for correlated sampling

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{U}(A)$ denote the uniform distribution on a finite set $A$, and let $(f,g)$ be probabilistic strategies satisfying the correctness condition for correlated sampling. For positive integers $n,a,b,\ell$ with $\ell\geq 2$ and $n\geq a+b-\ell$, consider sets $A,B\subseteq[n]$ with $|A|=a$, $|B|=b$, and $|A\cap B|=\ell$. Rivest's conjecture. For every such $n,a,b,\ell$ and every pair $(f,g)$, there exist sets $A,B\subseteq[n]$ with these sizes such that

$$\Pr[f(A)\ne g(B)]\geq 1-\frac{\ell}{a+b-\ell}=1-\frac{|A\cap B|}{|A\cup B|}.$$

This conjecture asserts that Broder's MinHash correlated-sampling strategy is optimal except in the case of intersection size one covered by Rivest's protocol. The paper's main result proves the asserted lower bound for constant-fraction intersections, while the fully general statement remains open.