  We address the issue of potential superluminal propagation of gravitational
waves in backgrounds neighboring the previously suggested bounce
[arXiv:1807.08361] in beyond Horndeski theory. We find that the bouncing
solution lies right at the boundary of the region where the gravitational waves
propagate at speed exceeding that of light, i.e. that solution suffers
superluminality problem. We suggest a novel version of a completely stable
bouncing model where both scalar and tensor perturbations remain safely
subluminal not only on the solution itself but also in its neighbourhood. The
model remains free of superluminality when extra matter in the form of
radiation or, more generally, ideal fluid with equation of state parameter
$w\leq 1/3$ (and also somewhat higher) is added. Superluminality reappears when
extra matter is added whose sound velocity is equal or close to 1 in flat
space; an example is scalar field minimally coupled to metric. The latter
property is characteristic of all beyond Horndeski cosmologies; we briefly
discuss its significance.
