Hu–Hwang–Wang cutoff-point conjecture for adaptive group testing

Let $S$ be a finite set of $n$ items, with $0<d<n$ defective items, and let $M(d,n)$ denote the minimum, over adaptive group-testing algorithms, of the worst-case number of tests needed to identify all defectives. Set $r:=n/d$.

Hu–Hwang–Wang cutoff-point conjecture. For $r\leq 3$,

$$M(d,n)=n-1.$$

This conjecture asserts that, up to the cutoff ratio $3$, testing items individually is optimal. The statement was proved for $r\leq \frac{21}{8}=2.625$ by Du and Hwang, so the full conjecture is resolved.