Both papers are about quantum theory in a classical curved background geometry, hence say nothing about quantum field theory, and hence nothing about locality.

Until someone is able to reconcile renormalization with Bohmian mechanics, the latter says nothing of interest about quantum field theory.

Bohmian mechanics cannot even produce QED, about which much more is known than about quantum gravity.

For QED, the paper https://arxiv.org/abs/0707.3487 mentioned by @RubenVerresen displays the problems with the Bohmian approach in quantum field theory. The authors propose a naive dynamics for the wave function in (23), then state immediately that the dynamics is not well-defined, and then go on to state wrong claims about what Symanzik nd Luescher showed. The same problems appear for their proposed equilibrium density in (24). They say the problems go away when one makes the number of degrees of freedom finite (which is true), but they remain silent about the fact that the problems reapppear when one tries to take the continuum limit.

Without the continuum limit one doesn't have Lorentz invariance (as they note), but then one only has a finite lattice approximation of QED. One doesn't have QED, which is the theory defined in the continuum by Lorentz covariance and minimal coupling.