Let $Γ$ and $ψ=\frac{\Gamma'}{\Gamma}$ be respectively the classical Euler gamma function and the psi function and let $γ=-ψ(1)=0.57721566\dotsc$ stand for the Euler-Mascheroni constant. In the paper, the authors simply confirm the logarithmically complete monotonicity of the power-exponential function $q(t)=t^{t[ψ(t)-\ln t]-γ}$ on the unit interval $(0,1)$, concisely deny that $q(t)$ is a Stieltjes function, surely point out fatal errors appeared in the paper [V. Krasniqi and A. Sh. Shabani, \emph{On a conjecture of a logarithmically completely monotonic function}, Aust. J. Math. Anal. Appl. \textbf{11} (2014), no.~1, Art.~5, 5~pages; Available online at \url{http://ajmaa.org/cgi-bin/paper.pl?string=v11n1/V11I1P5.tex}], and partially solve a conjecture posed in the article [B.-N. Guo, Y.-J. Zhang, and F. Qi, \textit{Refinements and sharpenings of some double inequalities for bounding the gamma function}, J. Inequal. Pure Appl. Math. \textbf{9} (2008), no.~1, Art.~17; Available online at \url{http://www.emis.de/journals/JIPAM/article953.html}].