Solved. The order of magnitude of the maximum is known:

$$\max_{A,B}F(A,B)\asymp\frac{N^2}{(\log N)^{\delta}(\log\log N)^{3/2}},\qquad \delta=1-\frac{1+\log\log2}{\log2}\approx0.086.$$

The upper bound is immediate from Ford's theorem on the number of distinct entries of the $N\times N$ multiplication table [Fo08] (claim page (Ford, 2004), partial). The lower bound is the result of a manuscript of 26 April 2026 credited to GPT-5.5 Pro prompted by Chojecki, which builds $A$ from multiples $pr$ of randomly chosen large prime labels $p$ and $B$ from the integers up to $N$ divisible by no chosen label, so that uniqueness within a label reduces to Ford's integers with exactly one divisor in a short interval; the site's curator, Thomas Bloom, accepted it on 2 May 2026 (claim page (Chojecki, 2026)). An earlier thread post of 23 November 2025 (van Doorn) first observed the upper bound from Ford's theorem and gave the weaker lower bound $(1+o(1))N^2/\log N$ from Szemerédi's construction; the site's commentary credited it until its edit of 2 May 2026. It is a thread post, not a dated manuscript, so it has no claim page; its upper bound is Ford's theorem, paged above. No leading constant is known or asked. The site's commentary was last edited on 2 May 2026.