  Let $1<p<\infty$. Let $\{T_t\}_{t>0}$ be a noncommutative symmetric diffusion
semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let
$\{P_t\}_{t>0}$ be its associated subordinated Poisson semigroup. The
celebrated noncommutative Littlewood-Paley-Stein inequality asserts that for
any $x\in L_p(\mathcal{M})$,
  \begin{equation*}
  \alpha_p^{-1}\|x\|_{p}\le \|x\|_{p,P}\le \beta_p \|x\|_{p},
  \end{equation*}
  where $\|\cdot\|_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions
associated with $\{P_t\}_{t>0}$, and $\alpha_p, \beta_p$ are the best constants
only depending on $p$.
  We show that as $p\to \infty$,
  $$ \beta_p\lesssim p, $$
  and $p$ is the optimal possible order of $\beta_p$ as well. We also obtain
some lower and upper bounds of $\alpha_p$ and $\beta_p$ in the other cases.
