  In this note two blow-up results are proved for a weakly coupled system of
semilinear wave equations with distinct scale-invariant lower order terms both
in the subcritical case and in the critical case, when the damping and the mass
terms make both equations in some sense "wave-like". In the proof of the
subcritical case an iteration argument is used. This approach is based on a
coupled system of nonlinear ordinary integral inequalities and lower bound
estimates for the spatial integral of the nonlinearities. In the critical case
we employ a test function type method, that has been developed recently by
Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar
solutions of the adjoint linear equation. Therefore, as critical curve in the p
- q plane of the exponents of the power nonlinearities for this weakly coupled
system we conjecture a shift of the critical curve for the corresponding weakly
coupled system of semilinear wave equations.
