Conjectured optimal strategy for nested group testing

Let $p\binom{}{}$ be the prevalence parameter, let $D_k(m,p)$ denote the cost of a nested strategy $(k,m)$, and let $m_{33}$ and $m_{34}$ be the strategies defined by

$$(1,m_{33})=(1,(3)),\qquad (k,m_{33})=(k,(3^k,\dots,3)),$$

and

$$(1,m_{34})=(1,(4)),\qquad (k,m_{34})=(k,(4\times 3^{k-1},3^{k-2},\dots,3)).$$

A strategy is optimal for $p$ when it minimizes $D_k(m,p)$ over all nested strategies. Conjectured optimal strategy. If $p\geq 1-3^{-1/3}$, then the optimal strategy is to test all individuals, with no pooling. If $p\leq 1-3^{-1/3}$, then there is a $k=k(p)\geq 1$ and an optimal strategy $(k,m)$ such that

$$(k,m)\in\{(k,m_{33}),(k,m_{34})\}.$$

The preceding theorem proves the same assertion with the additional possibilities $m_{23}$ and $m_{24}$; the conjecture claims that those two strategies are never needed. Its resolution would identify the optimal nested strategy throughout the full prevalence range.