Before revision:  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:  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(p\log(1/p)\big) the standard deviation of the number of tests per individual is of the same order as the cost .