A conjecture concerning a completely monotonic function

On the basis of a sequence of numerical computations, a conjecture is presented regarding the class of functions H(x;a)=exp(a)−(1+a/x)xH(x;a)=exp(a)−(1+a/x)x, and the open problem of determining the values of aa for which the functions are completely monotonic with respect to xx. The critical value of aa is determined here to sufficient accuracy to show that it is not a simple symbolic quantity.