The Laplace-transform criterion for $L_m(s)=1/(mse^s+1)$

For $m\ge 0$, define

$$L_m(s)=\frac{1}{mse^s+1}.$$

A Laplace transform of a random variable is understood in the usual probabilistic sense.

Laplace-transform criterion. The function $L_m$ is the Laplace transform of some random variable if and only if $m=0$, corresponding to a trivial random variable, or $m\ge e$.

For $m\ge e$, the associated inverse transform is known to give a probability density, while the conjectured extension to all $m\ge0$ is false because complete monotonicity fails for sufficiently small positive $m$. The supplied text gives no resolution of the sharper if-and-only-if claim.