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