We are interested in this article in studying the damped wave equation
with localized initial data, in the
\textit{scale-invariant case} with mass term and two
combined nonlinearities. More precisely, we consider the following
equation: \begin{displaymath} \d (E)
\hspace{1cm} u_{tt}-\Delta
u+\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=|u_t|^p+|u|^q,
\quad \mbox{in}\
\R^N\times[0,\infty),
\end{displaymath} with small initial data. Under some
assumptions on the mass and damping coefficients, $\nu$
and $\mu>0$, respectively, we show that
blow-up region and the lifespan bound of the solution of $(E)$ remain
the same as the ones obtained in [Our2] in the case of a
mass-free wave equation, {\it i.e.} $(E)$ with
$\nu=0$. Furthermore, using in part the computations
done for $(E)$, we enhance the result in [Palmieri] on the
Glassey conjecture for the solution of $(E)$ with omitting the
nonlinear term $|u|^q$. Indeed, the blow-up
region is extended from $p \in (1,
p_G(N+\sigma)]$, where $\sigma$ is
given by \eqref{sigma} below, to $p \in
(1, p_G(N+\mu)]$ yielding, hence, a better estimate
of the lifespan when
$(\mu-1)^2-4\nu^2<1$.
Otherwise, the two results coincide. Finally, we may conclude that the
mass term {\it has no influence} on the dynamics of
$(E)$ (resp. $(E)$ without the nonlinear term
$|u|^q$), and the conjecture we made in
[Our2] on the threshold between the blow-up and the global
existence regions obtained holds true here.