Before revision:  We conjecture that the optimal strategy, depending on the value of p, is indeed of the form (3^k , , 3 ) or of the form (3 ^{k-1}4,3^{k-1}  \dots,3  ), with a precise description for k. This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .
After revision:  We conjecture that the optimal strategy, depending on the value of p, is indeed of the form (3^k , , 3 ) or of the form (3 ^{k-1}4,3^{k-1} \dots,3 ), with a precise description of the range of p's where each holds. We then focus on schemes of the type (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost .