Optimality of the generalized pairwise testing algorithm for ordered probabilities

Let $u_1,u_2,\ldots,u_n$ be units ordered so that

$$p_1\leq p_2\leq \cdots \leq p_n,$$

with

$$1-1/\sqrt{2}\leq p_i\leq (3-\sqrt{5})/2$$

for all $i$. A nested testing procedure preserves this order if it tests units according to the fixed ordering $u_1,u_2,\ldots,u_n$. The generalized pairwise testing algorithm (GPTA) tests the first two units together and, after a positive joint test, individually tests the one with smaller contamination probability.

GPTA optimality conjecture. Among all nested testing procedures that preserve the order $u_1,u_2,\ldots,u_n$, the GPTA is an optimal nested procedure. It is not necessarily uniquely optimal at the boundary values of the interval.

This conjecture concerns optimal expected test counts in generalized group testing when all contamination probabilities lie in the specified interval. The source notes that the conjecture was proposed previously and that its first version was validated through Monte Carlo simulations for population sizes up to $1{,}000$; no proof or resolution is supplied here.