  Towards confirming Sun's conjecture on the strict log-concavity of
combinatorial sequence involving the n$th$ Bernoulli number, Chen, Guo and Wang
proposed a conjecture about the log-concavity of the function
$\theta(x)=\sqrt[x]{2\zeta(x)\Gamma(x+1)}$ for $x\in (6,\infty)$, where
$\zeta(x)$ is the Riemann zeta function and $\Gamma(x)$ is the Gamma function.
In this paper, we first prove this conjecture along the spirit of Zhu's
previous work. Second, we extend Chen et al.'s conjecture in the sense of
almost infinite log-monotonicity of combinatorial sequences, which was also
introduced by Chen et al. Furthermore, by using an analogue criterion to the
one of Chen, Guo and Wang, we deduce the almost infinite log-monotonicity of
the sequences $\frac{1}{\sqrt[n]{|B_{2n}|}}$, $T_n$ and
$\frac{1}{\sqrt[n]{T_n}}$, where $B_{2n}$ and $T_{n}$ are the $2n$th Bernoulli
number and the $n$th tangent number, respectively. These results can be seen as
extensions of some solved conjectures of Sun.
