Optimality of the Theorem uj1 algorithm for one defective search

Let $g(n,k,\alpha,m)$ denote the minimum number of group-testing queries in the model considered, and let $a$ be the query-size parameter. Assume that $\frac{1}{\alpha}$ is an integer and

$$k+\log k+1\leq \left\lceil \frac{n}{a}\right\rceil.$$

Optimality conjecture. Under these assumptions, the algorithm used in the proof of Theorem uj1 is optimal for $m=1$. The preceding discussion indicates that the algorithm seems optimal under the stated condition, while noting counterexamples when $1/\alpha$ is not an integer; the exact values of $g(n,k,\alpha,m)$ are otherwise difficult to determine.