  The paper by R. Garrappa, S. Rogosin, and F. Mainardi, entitled {\em On a
generalized three-parameter Wright function of the Le Roy type} and published
in [Fract. Calc. Appl. Anal. {\bf 20} (2017) 1196-1215], ends up leaving the
open question concerning the range of the parameters $\alpha, \beta$ and
$\gamma$ for which Mittag-Leffler functions of Le Roy type $F_{\alpha,
\beta}^{(\gamma)}$ are completely monotonic. Inspired by the 1948 seminal H.
Pollard's paper which provides the proof of the complete monotonicity of the
one parameter Mittag-Leffler function, the Pollard approach is used to find the
Laplace transform representation of $F_{\alpha, \beta}^{(\gamma)}$ for integer
$\gamma = n$ and rational $0 < \alpha \leq 1/n$. In this way it is possible to
show that Mittag-Leffler functions of Le Roy type are completely monotone for
$\alpha = 1/n$ and $\beta \geq (n+1)/(2n)$ as well as for rational $0 < \alpha
\leq 1/2$, $\beta = 1$ and $n=2$. For further integer values of $n$ the
complete monotonicity is tested numerically for rational $0< \alpha < 1/n$ and
various choices of $\beta$. The obtained results suggest that for the complete
monotonicity the condition $\beta \geq (n+1)/(2n)$ holds for any value of $n$.
