Understanding this spacetime diagram I am studying this paper which describes the metric of a Planck star, a state in the life of a massive star conjectured in the context of Loop Quantum Gravity: in a collapsing star, the pressure of quantum effect should be able to contrast and reverse the collapse of the star leading to an explosion; this stage is very brief in the star reference frame, but can last millions of year for an outside observer because the time slows down close to large gravitational fields: this is what we call black holes.
In the paper is reported the following spacetime diagram:

the light blue region is described by a Minkowski metric, the pink region is described by a Schwarzschild metric and the gray patch is the region in which quantum effects take place.
This diagram is not very clear to me  (and I suspect that I did not understand something in the spacetime diagrams in general).
Assume that I perturb the central object at time $t=0$: the central object might produce gravitational waves which are observed at infinity. Do these gravitational waves propagate along the upper thick line (which I think corresponds $u=0$)? This line is the boundary of all three regions: which metric should I use to study the GW propagation?
 A: That is a conformal diagram representing the Haggard-Rovelli spacetime. The thick line represents the collapsing/bouncing null shell. The region I is enclosed in the shell and for the Birkhoff theorem, it is Minkowski. The region II is a portion of the Kruskal extension of Schwarzschild. The peculiar thing about conformal diagrams is that the causal structure is preserved or in other words that null trajectories are always at $45^{o}$ degrees. The region III (and the corresponding symmetrical one above $t = 0$ ) is the one where the Einstein equations are no longer valid. The metric, due to spherical symmetry, has the same form in all the region
$$ds^{2} = -F(u,v)dudv + r^{2}(u,v)(d\theta ^{2} + \sin^{2}\theta d\phi^{2}) \equiv \bar{g}_{\mu \nu}dx^{\mu}dx^{\nu}$$
where $F(u,v)$ and $r(u,v)$ needs to be glued at the boundary of the regions (like in the eq.s (27)-(30) of the paper you linked).
The collapsing is spherically symmetric, so in general, there are not gravitational waves. If you instead perturb the quantum region you can draw gravitational waves as photon shells and study the GW propagation in the Regge-Wheeler approach
$$g_{\mu \nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$$
I think your question is more general and concerns the propagation of GW in general backgrounds. You can find a reference in these two papers: https://arxiv.org/abs/1503.08101 and https://doi.org/10.1103/PhysRev.108.1063.
