  We consider the Klein-Gordon equation on a star-shaped network composed of n
half-axes connected at their origins. We add a potential which is constant but
different on each branch. Exploiting a spectral theoretic solution formula from
a previous paper, we study the L-infinity-time decay via H\"ormander's version
of the stationary phase method. We analyze the coefficient c of the leading
term c t^{-1/2} of the asymptotic expansion of the solution with respect to
time. For two branches we prove that for an initial condition in an energy band
above the threshold of tunnel effect, this coefficient tends to zero on the
branch with the higher potential, as the potential difference tends to
infinity. At the same time the incline to the t-axis and the aperture of the
cone of t^{-1/2}-decay in the (t,x)-plane tend to zero.
