It appears that these theories are not relativistic and claim that energy is not conserved. Therefore they will have significant trouble explaining almost all particle physics experiments.
From the perspective of a theoretical particle physicist these theories are also awkward because they try to fiddle with non-relativistic quantum mechanics like the Schrodinger equation. But that equation merely arises in the non-relativistic limit of the more fundamental relativistic quantum mechanics, more commonly known as quantum field theory. Therefore, if you want to probe the foundations of quantum theory then quantum field theory seems like a much better place to start.
As an analogy, these attempts sound like a nineteenth-century person trying to understand gravity starting from F = m g h rather than F = - G m1 m2 / r^2.
For these reasons I would not be supportive of significant funding for theoretical or experimental research in these collapse theories.
Saying that Schroedinger's equation is a non-relativistic limit is a bit misleading. The typical choice of Hilbert spaces, sure, is non-relativistic. But QFT and any other quantum theory, is basically Schroedinger's equation on a sufficiently weird Hilbert space (from second quantization, graphs, grids, or strings)
If you (and the cited wikipedia article a few comments up) mean i d/dt | psi > = H | psi > then sure, that might refer to a relativistic theory and in that sense my comment was imprecise. But showing that a given Hamiltonian is relativistic generally requires work, and I still think there is merit to my point that these theories seem to start 'at the wrong end'.
Nobody is really expecting large objects to behave differently, but somebody should still check.