Capacity conjecture for nonadaptive group testing

Consider either combinatorial nonadaptive group testing, where $k$ uniformly randomly chosen items among $n$ are defective, or probabilistic nonadaptive group testing, where each of the $n$ items is independently defective with probability $p=k/n$. Define

$$T^*=T^*(n)=\max\left\{k\log_2\frac{n}{k},\;\frac{1}{(\ln 2)^2}k\ln k\right\}.$$

Consider an asymptotic regime in which $k=o(n)$, $k\to\infty$, and $T^*<(1-\delta)n$ for some $\delta>0$. The capacity conjecture. If $T>(1+\epsilon)T^*$ for some $\epsilon>0$, there exists a test design and decoding algorithm whose error probability tends to $0$ as $n\to\infty$. If $T<(1-\epsilon)T^*$, then for any test design and decoding algorithm the error probability is bounded away from $0$ as $n\to\infty$. In particular, when $k=\Theta(n^\theta)$ for some $\theta\in(0,1)$, the capacity of nonadaptive probabilistic and combinatorial group testing is

$$C=\min\left\{1,\;\ln 2\,\frac{1-\theta}{\theta}\right\}.$$

The first minimand is the trivial counting bound. Achievability is known in some parameter ranges for combinatorial testing and has been supported nonrigorously for probabilistic testing, but a matching design- and algorithm-independent converse for the second minimand remains missing in the source.